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<!-- #################################################################### --> <!-- ## ## --> <!-- ## examples.xml RCWA documentation Stefan Kohl ## --> <!-- ## ## --> <!-- #################################################################### --> < <Ignore=settings tester> <Example> <![CDATA[ gap> SizeScreen([75,24]);; gap> SetAssertionLevel(0); ]]> </Example> </Ignore> This chapter discusses a java.lang.StringIndexOutOfBoundsException: Range [77, 29) o andgroupsindetail.All ofthem showdifferent ofthepackage, and the order in which they appear is entirely are not ordered by degree of difficulty or interes they interest P <Index Key="Index Key="AssignGlobalsC><C<Index> <Index Key="AssignGlobals"><C>AssignGlobals</C></Index> The rcwa mappings, rcwa groups and other objects defined in this chapter can be found in the file <F>pkg/rcwa/examples/examples.g</F>. This file can be read into the current &GAP; session by the function <Ref Func="LoadRCWAExamples"/> which takes no arguments and returns the name of a variable which the record containing the examples got assigned to. Thevariableassignmentsmadein section of chaptercan be made be foun inthefile<>kgrcwaexamples/g<F> Thisfile canbe component of the examples record. The component names are given at the end of read intointo the G sessionby RefFuncLoadRCWAExamples">which no arguments java.lang.StringIndexOutOfBoundsExceptio It quite likely just a ewlittlemodifications or additional easy commands you can findglobal assignments a ofthischapter be -- made by applying the function <C>AssignGlobals</C> to the respective <!-- #################################################################### --> <Section Label the correspondingsections <P> <Heading> java.lang.StringIndexOutOfBoundsException: Range [0, 10) out of bounds for lengt </Heading> Thompson's group V, also known as Higman-Thompson group, is a finitely infinitesimple group. Thisgroup beenfound byGraham Higman cf. < additional commandsyoucan findout interestingthingsyourself <Example> <![CDATA[ gap> G := Groupjava.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for > java.lang.StringIndexOutOfBoundsException: Index 24 out of bounds for length 9 <),4)((4)24)(()3(4)> ] <.nbsp;Cite Key"">.Weshow isisomorphic toThompson's V. This!CDATA[[ We take a slightly different set of generators: <gap> G : Group(ist[0,1,4],[0,41][1,,,]] <![CDATA[ gap> k := ClassTransposition; gap> l := ClassTransposition(1,2,2,4);; gap> m := ClassTransposition(0(2)1(),()14)(14)2(4)),24),())java.lang.StringIndexOutO gap> isomorphismhas beenpointed John P McDermottjava.lang.StringIndexOutOfBoundsExcept gapjava.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0 ()(),1()24),02,(4),14),()) gap> G = H; # k, l, m and gap> :1,24);java.lang.StringIndexOutOfBoundsException: Index 39 ou java.lang.StringIndexOutOfBoundsException: Range [4, 5) out of bounds for length ] </Example> Now gap> G = H; # k, l, m and n generate G as well pagetrue ]] as /> <Now we verify that our four generators satisfy the relationsjava.lang.StringIndexOutOfBou <![CDATA[ gap> HigmanThompsonRels := > [ k^2, l^2, m^2, n^2, #nbsp;M\u/> Cn<C asnbsp;M\<M> >lk*m*kln*k*m*lkm, 2 >*kmnk*nm*n***, () " > (l*k*m*k*l*n)^3, (m*k*l*k*m*n)^3, # (4) " > l*m^*k(mn*l^2k #()" > (l*n*m*n)^5, # (6) " > (l*k*n*k*l*n)^3*k* l*kmkk***k**k*m, (2 java.lang.StringIndexOutOfBoundsException: I > ((l*k*m*n)^2*(m*k*l*n)^2)^3, # (> lk*m*k*l*)3,, m**l**mm*)3 #() java.lang.StringIndexOu > (l*m*n)^, #() > (> (l*k*n*k*l*n)^3*k*n*k*(m*k*n*k*m*n)^3*k*n*k*n,# (7) " >(**k**k*ll**nn*^, " > ( (l*lk*m*km*lnm*km*)4 9)" [IdentityMapping Integers ) Integers), IdentityMapping( Integers lm**ll*k*m*knk)^2 ()" dentityMapping (Integers, IdentityMapping( Integers ), (Integers),IdentityMapping(Integers) IdentityMapping() ) IdentityMapping( Integers ), IdentityMapping( Integers ), IdentityMapping( Integers ), IdentityMapping( IdentityMapping ,IdentityMapping(Intege IdentityMapping( Integers ), IdentityMapping( IntegersIdentityMapping(Integers),Iden ]]> </Example> We conclude that our group isIjava.lang.StringIndexOutOfBoundsException: Range [18, 1 But sinceThompson' group Vissimple group isis , this] In fact it is java.lang.StringIndexOutOfBoundsException: Index 20 out of bounds for < of all class transpositions which interchange residue classes sinceThompson' group V is simple and ofnbsp check<C></>containsall class transpositions which interchange java.lang.StringIndexOutOfBoundsException: Index 1 <Example> <![CDATA[CT([2,ntegers)C generated bytheset all gap>), > ct->Mod(ct) in [ 2 checkthat C><C> all class [ ), 1(4) ), ( 0(4), 2(4) ), ( 0(java.lang.StringIndexOutOfBoundsException: Index 0 out of bounds (1(4), 2(4) ), ), 24 ,( 1() () ,( 4,3(4 )] gap> ct>ct) in [2)java.lang.StringIndexOutOfBoundsException: Index 41 out of bounds f true ]]> </Example> Then we give a function which takes a class transposition <M>\tau \in {\rm CT(0(4),24) ) (0() ()) () 0)) (2,24 ,java.lang.StringIndexOutOfBoundsExcept afactorization ofan <>gamma</> <M>\tau^\gamma \in S</M> into <M>g_1 := \tau_{0(2),1(4)} \in S</M>, <>g_2: \au_0(2,() in S/M> <M>g_3 =\au_{(),(4) in/>, <M>g_4 := \tau_{1(2),2(4)} \in S</M>, <M>h_1 := \tau_{0(4),1(4)} \in S</M> java.lang.StringIndexOutOfBoundsException: Range [29, 3) out of bounds for lengt <Listing Type="GAP code"> <![CDATA[ ReducingConjugatorfactorization ofelement<M\gamma<M> local M>\tau^\g \S<M M: \0(),() in />, g1 := M>g_2 :=g_2 =\tau_{0(2)3() in <M, <>: tau_{(,(4 inS<M>, g2 := ClassTransposition(0,2,3,4); h2 := ClassTransposition(1,4,2,4); g3 =ClassTransposition12,,4)java.lang.StringIndexOutOfBoundsException: Index 36 o 4:=ClassTransposition1,,2,); F :=java.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0 w := One(F); if Mod(tau) <= 4 then return w; fi; # Before we can reduce the moduli of the interchanged residue classes, #we make surethat them haveatleast modulus 4java.lang.StringIndexOutOfBoundsException: : (); if Mod(cls[1]) = 2 then if Residue(cls[1]) = 0 then if (cls2])mod then =tau^g2;w : *F.java.lang.StringIndexOutOfBoundsException: Index 68 else tau := tau^g1; lse if Residue(cls[2]) mod java.lang.StringIndexOutOfBoundsException: Index 0 out of bo := g3w = .3 ; java.lang.StringIndexOutOfBoundsException: Range [0, 6) out of bounds for length i whileModtau)>4 do Nowwe cansuccessively reducethe if not: TransposedClassestau); cl -> IsEmpty(Intersection(cl,Support(tau)))) then Modcls] 2 then cls =TransposedClasses(au) h if Residue(ls[2] 4 1 : g2; w =w F.; hi->Length(Filtered(cls,cl->IsSubset(Support(hi),cl)))=1); h := h[1]; tau := tau^h; if h = h1 then w := w * F.5; else w := w * F.6; fi; fi; cl := TransposedClasses(tau)[2]; # class with larger modulus r := Residue(cl); if r mod 4 = 1 then tau := tau^g1; w := w * F.1; elif r mod 4 = 3 then tau := tau^g2; w := w * F.2; elif r mod 4 = 0 then tau := tau^g3; w := w * F.3; elif r mod 4 = 2 then tau := tau^g4; w := w * F.4; fi; od; return w; end; ]]> </Listing> After assigning <C>g1</C>, <C>g2</C>, <C>g3</C>, <C>g4</C>, <C>h1</C> and <C>h2</C> appropriately, we obtain for example: <Listing Type="Example"> <![CDATA[ gap> ReducingConjugator(ClassTransposition(3,16,34,256)); h2*g1*h1*g1*h1*g1*h1*g1*h2*g2*h2*g4*h2*g4*h2*g3 gap> gamma := h2*g1*h1*g1*h1*g1*h1*g1*h2*g2*h2*g4*h2*g4*h2*g3; <rcwa permutation of Z with modulus 256> gap> ct := ClassTransposition(3,16,34,256)^gamma;; gap> IsClassTransposition(ct);; gap> ct; ClassTransposition(1,4,2,4) ]]> </Listing> Thompson's group V can also be else : tau^g1;w: w .1;fi <Example> !C[ gap> x := Indeterminate(GF(2));; SetName(x,"x"); gap =PolynomialRing(F(),); gap> k := ClassTransposition(0,x,1,x);; gapap>l : (,,,^2;; gap> m := ClassTransposition(0,x,1,x^2);; gap> n := ClassTransposition(1,x^2,x,x^2);; gap> G := Group(k,l fh=h1 thenw :=w .5 w = * F6 fi; < overGF()[]with enerators ]]> </Examplecl := TransposedClassestau[2;#class larger Theofrepresentationcanlikewise simply checking the defining relations r mod 4 = 1 then tau := tau;w =w *F; Enter <C>AssignGlobals(e java.lang.StringIndexOutOfBoundsException: Range [15, 14) o toassign the lobal definedinthis section </Section> <!-- #################################################################### --> w <Section Label <Heading assigning <>C> Cg2</> <C>3<C>,<>g4</> <C></> <Ch2/C>appropriately, we obtainfor : </java.lang.StringIndexOutOfBoundsException: Index 4 out of bounds for length 0 1932 his notebookthe of integersjava.lang.StringIndexOutOfBoundsException: Index 16 out of bounds for <Example> <![CDATA[ gap> Collatz := RcwaMapping([[2,0,3],[4,-1,3],[4,1,3]]r permutation of Z with modulus 256> gap> Display(Collatz); Rcwa mappinggap>> IsClassTransposition(ct;; / | 2/3 if in 0) n - <(n-1/ n 1() | (4n+1Listing ShortCycles(Collatz,-50.50]50) : [ [ 0 ], [ -1 ], [ 1 ], [ 2, 3 ], [ -2, -3 ], [ 4, 5, 7, 9, 6 ], [ -4, -5, > 1xx,^); [105 ,, 62,83, 111 99 66 ] [ -44, -59, -79>n =(,x^,x,^2;java.lang.StringIndexOutOfBoundsException: Index 43 out ]> </Example java.lang.StringIndexOutOfBoundsException: Range [18, 3) out of bounds for lengt determined yet. In particular it defining relationsgiven </> containing Enter Nevertheless, the factorization routine included in this package can java.lang.StringIndexOutOfBoundsException: Range [33, 9) out of bounds for lengt i.e. java.lang.StringIndexOutOfBoundsException: Range [0, 16) out of bounds for l <Example> <![CDATA[ gap> Section Label":actoringTheCollatzPermutation" true java.lang.StringIndexOutOfBoundsException: Range [25, 3) out of bounds for lengt 212 ]]> </> SettingtheInfo levelof<C>/>equal &bsp; s <C>RCWAInfo(2);<java.lang.StringIndexOutOfBoundsException: Index 0 out of bounds fo java.lang.StringIndexOutOfBoundsException: Range [18, 11) out of bounds for leng space, this is not done in this manual. <P/> We would like to get a java.lang.StringIndexOutOfBoundsException: Range [0, 36) out of to factor the inverse -- just like the java.lang.StringIndexOutOfBoundsException: Index factorization routine, this has influence on decisions taken during the factoring process: <Example> <[[ java.lang.StringIndexOutOfBoundsException: Range [9, 10) out of bounds for lengt 129 ]]> <[ ,[- ] [ ,[2 ,[- 3 ][ , 5 , ,6, This is already a shorter product, [-, -5,-79 - ] java.lang.StringIndexOutOfBoundsException: Range [2, 1) out of bounds for length looks very similar to Collatz' permutation. Therefore it is straightforward to try to factor both <C>mKnot(3)</C> [-44 59,-79 -,-0 93, -,-3,-111,-,-99] java.lang.StringInd lookwhetherthe of numbers of factors is ess nbsp129: <Example> <![CDATA[ gap> KnotFacts := Factorization(mKnot(determinedyet. Inparticularit not whetherthecycle gap QuotFacts =Factorization(mKnot()); gap> List([KnotFacts,QuotFacts],Length); [ 59, 9 ] gap(QuotFacts,KnotFacts) [ (determineafactorizationof java.lang.StringIndexOutOfBoundsException: Range [34, (6,2()) ( (6)),(0(),4()), (),1()) ( 0(3), 1(6) ), ( 0(36), 35(36) ), ( 0(36), 22(36) ), (36, (36)), 0(),17(6) ,(0(36) 1436 , ( 0(36), 20(> Collatzin CT(Integers) # ` in thegroup CT(Z. ( 2(36), 16(36) ), ( 2(36), 13(36) ), ( 2(36), 9(36) ), (236) () ) 2(), 636 ) (36) 336 ,java.lang.StringIndexOutOfBoundsException: Index 56 out ) ) (236),1536 ,(236,()) ( 2(36), 5(36C>RCWAInfo()</)causes the to displaydetailed (21() 30),( 2136, 23()) 2136) 34()) ( 21(36), done this .<> ( 21(36), Wwould to get factors ,we try (10(8, ()) (8,35(6) ) (10(18) 36 , 5) (36)), (12) () ),(6(),(18 ) ( 3(9), 17process: ( 0(9), 16(18) ), ( 6(9), 11(18) ), ( 3(9), 11(18) ), ( 0(9) ( 6(9), 4(18) )]]> ( 0(6), 2(24) ), ( 8(12), 17(/Examplejava.lang.StringIndexOutOfBoundsException: Index ( 8(12), 11(18W remember <>mKnot/C>,of whichthe permutation <C>mKnot(3)</C> (looksvery Collatz'permutation. Therefore is straightforward try to factor <>()/> <C>Collatz/)<C> to gap> look whether the sum of the numbers of factors is less than 129: rue ]]> </Example> The factors <C>PrimeSwitch(3gap>QuotFacts:=FactorizationCollatz//mKnot(3);; (cf. <Ref Func="PrimeSwitch>List(K,],Length); Enter <C>AssignGlobals59, 9] java.lang.StringIndexOutOfBoundsException: Range [34, 2) out of bounds for lengt <Section <!-- #################################################################### --> <Label":nplus1Group" <Heading> The <M( 036) 1836)) (36),(36 ) (),(36)) </> The following group acts transitively on the set of positive integers forwhich the <>n1<M> which by6 <Example> <![CDATA[ gap> :=ClassTransposition,2,,);; gap> b := ClassTransposition(1,3,2,6);; gap> c := ClassTransposition(2,3,4,6);; ap> =Group(,,) <(1(2),4(6)),(1(3),2(6)),(2(3),4((() ()) (2136, 2836) ,( 21(36)33() ), gap> LoadDatabaseOfGroupsGeneratedBy3ClassTranspositions(); "3CTsGroups6" gap> 3CTsGroups6 36 36) 21) 36 ,(21() ()) 44132 ]]> <Example> othis,consider the action MGM onthe <Q<M>3n1/ <>. The vertices of this tree are the positive integers for which java.lang.StringIndexOutOfBou conjecture holds, and for every vertex <M>n</M> there is an edge from <</M> denotes Collatzmapping <Alt Only="LaTeX"> <Display> <![CDATA[T: \ \ \mathbb{Z} \longrightarrow \mathbb{Z}, \ \ \ \ n 0() (18 , ( () ()) (39,11() )(0() () , \begin{cases} \frac{n}{2} & \text{if} \ \ n \ (),4)) () 418 ),(0(9) (18 ) (6, 14()) \( 0(6),(24)) (8(2) (8 , (7(12) 17(8), end{}]java.lang.StringIndexOutOfBoundsException: Index 18 out of bounds for leng / </Alt> <Alt Only="HTML"><![CDATA[<center> <img ="ollatz..png width ""height "" alt = "T: Z -> Z, n |-> (n/2 if n even </center>]]></Alt> <Alt Only="Text"/Example> / /2 neven, unc= =p/)<> | (3n+ <>java.lang.StringIndexOutOfBoundsException: Range [23, 22) out of bounds for \ ]]></Verb> (;<Ref Label"h:/)java.lang.StringIndexOutOfBoundsException: Index 47 out of bounds It is easy to check that for every vertexHeading> <M>b</M> or <M>c</M> maps <M>n</M> to <M>T(java.lang.StringIndexOutOfBoundsException: Index 0 out of other two generatorsfor which <M3+</> conjectureholds which java.lang.StringIndexOutOfBoun <Example So>3+/M>conjectureisequivalentto theassertionthat java.lang.StringIndexOutOfBoundsEx group <MG<M>acts transitivelyonM{} setminus06</. First let's have a look at balls of small radius about 1 under the actionof MG/ -these of those numberswhose under <M>gap>c: (,34); <Example> <![CDATA12),(6),((),())((),6> gap> Ball(G,1,5,OnPoints); [ 1, 2, 4, 5, 8, (); gap> Ball(gap> CTsGroups6.d3CTsGroup(G,3CTsGroups6.rps; 'catalogue number' of G [ 1, 2, 3, 4, 5, 44132 128, 160, 170, 256, 320, 340, 341, 512, 1024, 2048 ] gap> seethis,consider theactionof <><M the QM>3+11<M> tree</> [ 1, of treearethepositive for the<>n+</> 35, 40, 44 holds andfor vertexM><M there is anedgefrom ,136 ,141, 151,,170,208,212,213,226,227,256,272, 277, 280, 301, 302, 320, 340, 341, 416, 424, 452, 453, 454, 512, 640, ,832,853,904,908 909 , , ,, , 1664, 1696, 1706, 1808, 1813, 1816, 2048, 2560, 2720, 2728, 4096, ,5456,5461,8192 ,10880,10912,10922, , 32768, 65536 ] gap>\frac3+1}2} &\{}\ \n\\ tis odd} [ [ 1 ], [\ndcases}]> [ 3/ [ 26,/ [ 17, 104, 106, 113, 640, 680, 682, 4096 ], [ 34, 35, 208, 212, 213, 226, 227, 1280, 1360, 1364, 1365, 8192 ], , , ,70, 75, 416, 424, 452 454 2560 ,16384 ], [ 22, 23[11 68,69 ,416,424,452,453,454,2560 2720 5440, 5456, 5461, 32768 ] 22,23 , 151 ,java.lang.StringIndexOutOfBoundsException: Index ,45,46, 272 277,280,301,302,1664,1696 1706 1808 1813, 1816, 10240, 10880, 10912, 10922, 65536 ] ] ((,,50,,Length; [ 1, 2, 1, 1, 2, 2, 4, 2, 4, 4, 6, 8, 12, 14, 17, 20, 26, 32, 43, 52, 66, 81, 104 ,28924 36532, ,58399, 1730 ,,35224429, , ,8978 ,14296 .7457 1., 22828, 28924, 36532, 126056, 125752, 1.126537 ., ., gap> 1.26315, 1.26415, 1 126303,1.6317 1., . ] [ 2., 0.5, 1., 2., 1., 2>Difference(Filtered(1.100,n-n mod6 < ),BallG140OnPoints)java.lang. 1.21429, 1.17647, 1.3, 1.23077, 1.34375, 1.2093, 1.26923, 1.22727, >List ., 27259 25544,1.,1., .27457, .26712, 1.26056, 1.25752, 1.26688, 1.26537, 1.26451, 1.26342, 1.26034,toM<Mjava.lang.StringInde 1.26315, 1.26415, 1.26704, 1.26303, 1.26317, 1.26553, 1.java.lang.StringIndexOutOfBoun gap> Difference[a ,c] - 1() 46 ,(3) () , ( (),4()) java.lang.StringIndexOutOfBoundsException [ 27, 31, 41, 47, 55, 62, 63, 71,/Examplejava.lang.StringIndexOutOfBoundsException: Inde gap> T := RcwaMapping([[1,0,2],[3,1,2]]);; gap> List(last2,n->java.lang.StringIndexOutOfBoundsException: Index 24 out of bound [ 71, 68, 70, 67, 72, 69, 69, 66, 74, 71, 71, 60, 68, 68gap> List(B,Length); ]]> </Example> convenient anepimorphism fromthefreegroupof rank to <M>G</M>: <Example> <![CDATA[ gap> F := FreeGroup("a","b","c"); f group [a,b, > gap> phi := EpimorphismByGenerators(F,G); [ a, b, cgap> g =a*b;; Orderg);; ]]> </Example> We can compute <Example> <![CDATA[ gap> B := (6),(6,418,212 )(() 36,10)) gap> List(B,Length); [ , 6 ,24 48 192 ] gap> List(B[3],Order); [ 12, infinity, infinity, infinity, infinity, 12 </Example gap> List(B[3],g->Spending some more time to compute <C>B := Ball(G,One(G),12:Spheres);;</C>, [b*a,c* *aa ca*, * ] gap> g := a*b;; Order(g);; gap> Display(g); Rcwa permutation of Z with modulus 18,3 ,but which identityin M/> (1(), 836), (18) () )(36) () (18 java.lang.StringIndexOutOfBoundsException: Index 54 ou ( 5(6), 32(36), 16(18) ) ]]>[[ gap> =(*)^3*b*; Orderg; Spending some more time to compute <C>B := java.lang.StringIndexOutOfBoundsException: Rang one cancheckthat<>(){12}/>isthe shortest generators of <M>G</M> which 3 cyclic groups of order 2,288) 96,128(),32(36)) owever, thegroup <><M> ofother orders as well -- for example: <> <![CDATA[ gap> g := (b*a)^3*b*c;; Order(g);; gap>Order(*cb*****)java.lang.StringIndexOutOfBoundsException: Range [28, 29) out Rcwa permutation of Z <> ( 8(9),1618, ()256(,85(6,128144,32) java.lang.StringIndexOutOfBoundsException: Inde ( 712,11(),2236))( 5) 10() 40(144) 13(48, 20(72) ) ( 1(24), 2(36), 4(72) ) ( 14(36), 28(72), 112(288), 37(96), 56(144) ) gap> Order(a*c*b*a*b*c*a*c); 60 ]]> </Example> With some more efforts, one finds that e.g. <M>(abc)^2c^b</M> has order 616, that <M>(abc)^2b</M> has order 2310, that <M>(ab)^2a^ca^bc</M> has order 27720, and that <M>a(c(ab)^2)^2</M> has order 65520. Of course <M>G</M> has many elements of infinite order as well. Some of them have infinite cycles, like e.g. <Example> <![CDATA[ gap> g := b*c;; java.lang.StringIndexOutOfBoundsException: Index 2 out of bounds for length 0 Rcwa permutation of Z with modulus 12 / | 4nRcwa permutation of Z with modulus 12 | 2n if n in 5(6) n |-> < n/2 if n in 2(12) | n/4 if | n nin1(3) n nin 03 \ gap> Sinks(g); [ 4(12) ] gap> Trajectory(g,last[1],10); [ 4(12), 16(48), 64(192), 256(768), 1024(3072), 4096(12288), 16384(49152), 65536(196608), 262144(786432), 1048576(3145728) ] gap> Trajectory(g,4,20); [ 4, 16, 64, 256, 1024, 4096, 16384, 65536, 262144, 1048576, 4194304, 16777216, 67108864, 268435456, 1073741824, 4294967296, 17179869184, 68719476736, 274877906944, 1099511627776 ] ]]> </Example> Others seem to have <Q>on average</Q> comparatively <Q>short</Q> cycles, like e.g. <Example> <![CDATA[ gap> g := a*b*a*c*b* <rcwa of modulus 144> gap> : (g,010000,,020;java.lang.StringIndexOutOfBoundsException: Index 50 out of bo gap> Difference([0..10000],Union(cycs)); [ ] gap> Collected(List(cycs,Length)); [ [ 1, 2222 ], [ 3, 1945 ], [ 4, 1111 ], [ 5, 93 ], [ 6, 926 ], java.lang.StringIndexOutOfBoundsException: Range [28, 2) out of bounds for lengt [ 13, 1 ], [ 14, 31 ], [ 16, 12 ], [ 18, 4 ], [ 20, 1 ] ] ]]> </Example> Iffthecycle of <g<M containing <>nn\in &ZZ;<M>isfinite and has a , 16, 64,256,1024 , 16384 ,java.lang.StringIndexOutOfBoundsException: Index 70 such that for every <M>k \in &ZZ;</M> the cycle <M>n + km</M> has length <M>l</M> as well. Thus, in other words, every <M><Qbelongsto</>a cycle ofresidueclassesjava.lang.StringIndexOutOfBoundsException: Ran (This is a special property of <M>g</M> which is not shared by every rcwa permutation -- cf. e.g. Collatzjava.lang.StringIndexOutOfBoundsException: Index Section <Ref Label="sec:FactoringTheCollatzPermutationrcwa permutation of Z with modulus We can find some of these infinitely many <Q>residue gap>cycs = ShortCycles(g,0.],,10^); < <![CDATA[ gap> cycsrc := ShortResidueClassCycles(g,Mod(g),20); [ java.lang.StringIndexOutOfBoundsException: Index 4 out of bounds for length 4 [ 7(18), 352(864), 44( ,2222 ,[3 1945 ] [ 4 ,[5 ,[6,926 ] [ 11(18), 544(864), 2896(4608), 362(576), 68(108), 884 ], [ 12, 95 ], () () (108) 52()],[10()] [34), [1(4,642592),8() 4216,16(1152) ()] [ 5(54), >has <>/>aswell Thus inother every (2304 () , [ 17(54), 832(2592), 4432(13824),his a pecialproperty of<M>g<>whichisnotshared 104324,136()], [ 37(Sectionnbsp;Ref(6), 160,2036 , [ 41(54[7(8,352864) (),28(2)] () 288)] [ 53(54), 2560([13(8) 640(864),108, 3434()] 1706(1728), 320(324), 424(432) ], [ 38(72[,642592,8() (216,(1152,2), 6272,94108) ()]] gap> List(cycsrc,Length); [ 1, 3, 4, 6, 4, 1, 1, 6, 8, 8, 6, 8, 8, 3, 3 ] gap> Sum(List(Flat(cycsrc),cl->1/Mod(cl))); 97459/110592 gap>Float(ast); about%'' 0.881248 gap : (,3*Modg),; [ [ 0(6) ], [ 3(6), 160(288), 20(36) ], [ 7(18), 352(864), 44(108), 28(72) ], [ 11(18), 544(864), 2896(4608), 362(576), 68(108), 88(144) ], [13(),640864,(108,52(72 , () ,36 , [ 1(54), 64(2592), 8(324), 4(216), 16(1152), 2(144) ], [ 5(54), 256(2592), 1360(13824), 170(1728), 32(324), [ 7(18), 352(864), 44(108), 28(72) ], 208(2304), 26(288) ], [ 17(54), 832(2592), 4432(13824) (), (),5272) ][ 1036],[ 34(34()] (324,136432) ,java.lang.StringIndexOutOfBoundsException: Index 28 out of bounds f [ 37(54), 1792(2592), 224(324), 148(216), 784(1152), 98(144) ], [ 41(54), 1984(2592), 10576(13824), 1322(1728), 248(324), 328(432), 1744(2304), 218(288) ], [53(4) () 1364813824 (,9098() 1706(1728), 320(324), 424(432) ], [ 38(72), 58(108), 304(576) ], [ 62(72), 94(108), 496(576) ], [ 23(162), 1120(7776), 5968(41472), 746(5184), 140(972), 184(1296), [41) (2592 1057613824 976(6912), 5200(36864), 650(4608), 122(864) ], [ 35(162), 1696(7776), 9040(41472), 48208(221184), 257104(1179648), (972),280(296)], [ 73(162), 3520(7776), 440(972), 292(648), 1552(3456), 8272(18432), 1034(2304), 194(432) ], (162),3712, 1979241472),2474(),464() (), 3280(6912), 17488(36864), 2186(4608), 410(864) ], [ 89(162), 4288(7776), 22864(41472), 121936(221184), 650320(1179648), (,15242(27648, 28585184, 972 ()] [ 127(162), 6112(7776), 764(972), 508(648),[35162, 1696(776) (),48208(221184,257104(17 18022304) 338()] [ 14(216), 22(324), 112(1728), 592(9216), 74(1152) ], [ 86(216), 130(324), 688(1728), 3664(9216), 458(1152) ] ] gap) [ 1, 3, 4, 6, 4, 1, 1[ 7162) 1979241472 (5184,464972, (1296) gap> Sum(List(Flat(cycsrc),Density)); 5097073/5308416 gap> Float(last); # already about 96% 'coverage' 0.960187 ]]> </Example> There are 1802(2304,338432],java.lang.StringIndexOutOfBoundsException: Index 30 ou all finite, but <Q>on average</Q> pretty <Q>long</Q> -- e.g. <Example> ![CDATA gap> g [,3,46, 1 , 6 8,8, 6,8 8 3 10,108, , 10 ,5 gap> Display(5] Rcwa permutation of Z with modulus 288 gap>() #already about 96 coverage 16n1/ n 13 < | (27n+19)/4 if n in 15(24) U 23(24) (n+java.lang.StringIndexOutOfBoundsException: Index 9 out of bounds for length 9 | Display()java.lang.StringIndexOutOfBoundsException: Index 16 out of bounds for | (27n+29)/8 if n in 9(48) U 41(48) | (9n+7)/8 if n in 17(48) U 33(48) 2-7/ if n (36 n |-> < (4n-11)/9 if n in 32(36|(3n+)4if n 524) | (27n+38)/8 if n in 14(48) | (3n+2)/8 if n in 26(48) | (9n+10)/8 if n in 38(48) 34/ n in20) | n/4 if n in 56(72) | (|(2-)9 if n (36 | (27n+58)/16 if n in 50(96) | n if n in 0(6) \ gap> List([1..100],n->Length(Cycle(g,n))); [ ,,6,6,1,194 ,216, , 26 1,26, , 65 , , 1 216 26, 6, 216, 46, 1, 640, 26, 70, 194, 216, 1, 70, 26, 216, 216, 26, 1, 194, 216, 73, 26, 110, 1, 194, 216, 194, |()16if in () 194, 26, 1, 171, 194, 204, 640, 216, 1, 111, |(n+58)16 if in 50) ,26,111 ,1,50 ,26,216 640 26 111 40,110 1, 26, 194, 385, 640, 88, 1, 100, 111, 65, 110, \ ] gap> Length(Cycle(g[6,1 6 66 6 1 194,6,216,26,26,1,26,194, ,,26 1,216, 640 194,216,73 ,110,1,194 216 , ,39 1, 194, , 323270249684063829 gap> Length(Cycle(g,25855)); 4751 gap> Maximum(Cycle(g,25855)); 507359605810239426786254778159924369135184044618585904603866210104085 gap> cycs := ShortCycles(g,[0..50000],10000,10^100);; java.lang.StringIndexOutOfBoundsException: Index 34 out of bounds for length 22 gap> for cyc in cycs do S := java.lang.StringIndexOutOfBoundsException: Index 34 out of b gap> S;gap>Maximum(ycleg,))java.lang.StringIndexOutOfBoundsException: Index 26 out [ ] ]]> </Example> Taking a look at the lengths of the trajectories of the Collatz mapping <M>T</M> starting at the points in a cycle, we can see how a cycle of <M>g</M> goes <Q>up and down<gap> do DifferenceScyc) java.lang.StringIndexOutOfBoundsException: <Example> <![CDATA[ gap> List(Cycle(g,25),n->Length(Trajectory(T,]> [ 17, 21, 25, 29, 33, 31, 35, 34, 32, 33, 37, 41, 45, 44, 42, 39, 43java.lang.StringIndexOutOfBou 41, 45, 44, 42, 43, 40, 38, 35, 39, 37, 41, 40, 44<[CDATA[ ,45 46 ,48 ,45,46,50 ,4,43 ,38,39, 36, 34, 30, 27, 31, 29, 33, 32, 30, 31, 35, 33, 37, 36, 40, 39, 43, 41, 45, 44, 42, 43, 47, 51, 55, 53, 57, 56, 54, 55, 59, 58, 62, 66, 64, 68, 67, 65, 47,48,45,,,44 ,47,45,46,50 ,47,43,41,38,39, 36,34,30,27,31,29,33 50, 48, 45, 49, 47, 51, 50, 54, 52, 56, 55, 53, 54, 58, 62, 66, 70, 74, 72, 76, 75, 79, 83, 87, 91, 90, 94, 93, 97, 95, 99, 98, 96, 97, ,91 ,85,89 ,91 90 88 89 ,84,81,85, 83,87,86, 90, 94, 98, 97, 101, 105, 109, 107, 111, 110, 108, 109, 113, 117, 115, 119, 118, 122, 126, 125, 123, 120, 124, 122, 126, 125, 123, 124, 121, 119, 116, 117, 114, 111, 115, 113, 117, 116, 114, 115, 119, 123, 122, 120, 117, 121, 119, 123, 122, 120, 121, 118, 116, 112, 110, 106, 103, ,105 ,108,106, ,111, ,113 ,116 ,118,117 115, 116, 113, 110, 111, 108, 104, 102, 99, 103, 101, 105, 104, 108, ,110,109 107 ,112 111 109, 102,103,100,98,95,, 92, 96, 94, 98, 97, 95, 96, 93, 91, 88, 92, 90, 94, 93, 97, 101, 105, 09 ,106,103,107, 105,, ,106,107 104,102,99103 101, 105, 104, 108, 112, 110, 114, 113, 111, 112, 116, 115, 113, 109, 106, 110, 108, 112, 111,93 ,88,92,90 94,93 ,101,105, 105, ,103,100,98,95, ,97 101,100 , 107 , 109, 108, 106, 107, 104, 101, 98, 99, 96, 94, 91, 92, 89, 87, 84, 85, ,80 , 79,83 82,86 , 89,88 86,83,80 81,78,76, 73, 74, 71, 68, 72, 70, 74, 73, 71, 72, 76, 80, 79, 83, 87, 91, 90, 88, 85, 89, ,80,77,81, 79,83 ,85,89 88 86 83 80, 81,78 76, 92 ,95,93 94 98 96,100,99,97 98 102,106 114,113, 111, 108, 112, 110, 114, 113, 111, 112, 109, 107, 104, 108, 106, 110, 109, 113, 117, 115, 119, 118, 116, 117, 114, 111, 115, 113, 117, 116, 114, 115, 119, 118, 116, 112, 110, 107, 108, 105 111 ,112,110,114,113 , 112,109 107,104,108,10 106, 105, 109, 108, 112, 110, 114, 113, 111, 112, 116, 115, 113, 109, 106, 103, 104, 101, 99, 95, 91, 88, 92, 90, 94, 93, 91, 92, 96, 94, 98, 97, 95, 96, 100, 98, 102, 101, 105, 104, 102, 99, 100, 97, 93, 89, 87, 84, 85, 82, 80, 77, 74, 78, 76, 80, 79, 77, 78, 75, 73, 69, 67, 64, 68, 66, 70, 69, 73, 71, 75, 74, 72, 73, 70, 67, 68, 65, 63, 60, 64, 62, 66, 65, 69, 68, 66, 63, 64, 61, 59, 56, 60, 58, 62, 61, 65, 64, 62, 59, 60, 57, 55, 51, 48, 49, 46, 44, 40, 37, 34, 35, 32, 28, 26, 23, 27, 25, 29, 28, 32, 30, 34, 33, 31, 32, 36, 35, 33, 29, 26, 27 ,22, , 18,14, , 22, 20, 24, 23, 21, 22, 19, 16, 20, 18, 22, 21, 19, 20, 24, 23, 21, 17, 15, 17, 15, 19, 18, 16 ] gap> lngs := List27 24 22, ,19,23 21,,24,28,27 25 22 23 ,18 14,18, gap> Minimum(lngs); gap> 521 gap> Position(lngs,55); 15 gap> Position(lngs,521); 2807 ]]> </Example> Finally let's have a look at elements of <M>G</M> with small modulus: <Example> <![CDATA[ gap> B := RestrictedBall(G,One(G),20,36:Spheres);; gap> List(B,Length); [ 1, 3, 6, 12, 4, 6, 6, 4, 4, 4, 6, 6, 3, 3, 2, 0, 0, 0, 0, 0, 0 ] gap> Sum(last); 70 gap> Position(last2,0)-2; 14 ] </,401 ,405 ,,, 338,330 332, wehave 70elementsof 36or lessinMG/>which can be reached from the java.lang.StringIndexOutOfBoundsException: Index 29 out of bo generators without passing elements with mudulus exceeding 36. java.lang.StringIndexOutOfBoundsException: Range [15, 7) out of bounds for lengt element with;14. Now we double our bound[48, 4,[ ]] <Example/Example> <![CDATA[ gapWe there are elementsin MG<M> with modulus72 ListB,ength); [ 1, 3, 6, 12, 22, 14, 18, 22, 24, 26, 26, 34, 35, 32, 37, 38, 46, 58, ,82,91 , ,110, 114 ,146,138,148 168 74, 196, 215, 214, 232, 255, 2800212,((),(),()06))> ,,405, , 405 , ,415,435,424,401 ,,332 281 =ShortOrbitsG[..]100);java.lang.StringIndexOutOfBoundsException: Index 43 out ,88,55,35,20 , 5,1 2,0, ,0,0 0 , 0 ,0 0 ,, 8 ,8 ,2 2 0, 0, 0, 0, 0 ] gap Sumlast; 15614 java.lang.StringIndexOutOfBoundsException: Range [25, 26) out of bounds for leng 83 gap> Collected(List(Flat(B),Modulus)); [ [ 1, 1 ], [ 6, 3 ], [ 12, 4 ], [ 18, 2 ], [ 24,gap Collected(ast)java.lang.StringIndexOutOfBounds [484] 72,15540 ]] ]]> </Example> We observe that there are 15540 elements in <M>G</M> with modulus 72 which <Q>reachable</>from the identity successive withw elements with mudulus exceeding 72. Further we see that the longest word in the generators yielding an element So almost half of the inin range <C[.1000<C> java.lang.StringIndexOutOfBou Itis obviousmanyquestionsregarding group<M><M remain open. </Section> <!-- #################################################################### --> <java.lang.StringIndexOutOfBoundsException: Range [9, 8) out of bounds for length <Heading> huge orbits </Heading> In this section we investigate a group which has huge finite orbits on &ZZ;. java.lang.StringIndexOutOfBoundsException: Index 68 out of bounds for length 9 < Position(,[)java.lang.StringIndexOutOfBoundsException: Index 20 out of bounds fo gap> a := ClassTransposition(0,2,1,2);; gap(0,545;java.lang.StringIndexOutOfBoundsException: Index 39 out of bounds for gap> FlatB)java.lang.StringIndexOutOfBoundsException: Index 22 out of bounds for l gap> G := <(0(2),1(2)),(0(5),4(5)),(1(4),0(6))> gap> LoadDatabaseOfGroupsGeneratedBy3ClassTranspositions :=ab*;java.lang.StringInde "3CTsGroups6" gap> 3CTsGroups6.Rcwa permutation of Z with modulus 60 1284 ]]> </Example> We look for orbits of length at most | (0) range<C>[0.1000/> <Example> <![CDATA[ gap> orbs := ShortOrbits(G,[0..1000],100);; gap> List(orbs,Length); [ 16, 2, 24, 2, 2, 2, 8, |n3 nin1()U 8, 2, 2, 8, 2, 24, 8, 2, 56, 2, 2, 2, 8, 2, 8, 2, 2, 8, 2, 8, 2, 2, 2, 24, 2, 8, 2, 8, 2, 2, 8, 2, 8, 2, 24, 2, 2, 2, 8, 2, 8, 2, 2, 8, 2, 8, ,2 2, 2, 2 ,24 ,8,2 8 ,24,8 ,2, ,2, , ,8, ,2, 8, 2, 8, 2, 2, 2, 24, 2, 8, 2, 8, 736:after250000 stepstheiteratehas 414binary java.lang.Strin gap [166,2 ,[ ],[176 24,[,2 ] [196 ] [ [ 2, 67 ], [ 8, 32 ], [ 16, 1 ], [ 24, 9 ], [ 40, 1 ], [ 56, 1 ] ] gap> Length(Difference([0..1000],Union(orbs))); 491 ]]> < ,8 ,[ 322 2 ][326 2] 332,8], [ 346 ,java.lang.StringIndexOutOfBoundsException: Index 62 So almost half of the[386,2 , 392,24 ,[406,2 ,[412 8 ][ 416 200 ] orbits of length larger than 100. In fact there are much larger orbits. For example: < <![CDATA[ gap> B := Ball(G,32,500,OnPoints:Spheres);; # compute ball about 32 PositionB,];#>fail > exhausted orbit 354 gap> L(BL); #theorbit 6296 > Maximum(Flat()# largest integerin orbit 3301636381609509797437679 gap> B := Ball(G,736,5000,OnPoints:Spheres);; # the same for 736 ... gap> Position(B,[]); 2997 gap> Sum(List(B,Length)); # the orbit length for this time 495448 gap> Maximum(Flat(B)); java.lang.StringIndexOutOfBoundsException: Range [74, 73) out of bounds for leng [ 802, 2 ], [ 806, 2 ], [ 812, 8 ], [ 826, 2 ], [ 832 120 , 847078493406287854573381920553713155967741550498839 ]]> <java.lang.StringIndexOutOfBoundsException: Range [10, 9) out of bounds for lengt It seems that the cycles of <M>abc</M> completely traverse [952 , [ 956,956 8, [962,2] ,24,[,2, of <M>G</M>, with the only java.lang.StringIndexOutOfBoundsException: Index 29 out of bou in the above examples: <Example> <![CDATA[ gap> g := a*b*cthan 10000001000000 thecycle containing has length245719352 Displayg; Rcwa permutation of Z with modulus 60 / | n-1 if checking his mayrequire a of computing <P/ | 29(30) | 3n/2 if n in 0(20) U 12(20) U 16(20) | n+1 if n in 2(20) U 6(20) U 10(20) | (2n+1)/3 if n in 7(30) U 13(30) U 19(30) | n+3 if n in 1(30) U 11(30) n |-> < n-5 if n in 15(30) U 25(30) | (3n+12)/2 if n in 4(20) | (3n-12)/2 if n in 8(20) + if in (20java.lang.StringIndexOutOfBoundsException: Index 34 out of bounds for le | n-3 if n in 18(20) |(n7/ n 5(0 | (2n+9)/3 if n in 21(those here: \ gapCDATA[ 6296 gap> Length(Cycle(gap> h := a*b*c*a*c 495448 ]]> <Example Representatives and lengths of the cycles of <M>g</M> which intersect nontrivially with the range <C>[0..1000]</C> ,247724] <Example> !CDATA[ gap> CycleRepresentativesAndLengths(g,[0..1000]:notify:=50000); n = 736: after 50000 steps, the iterate has 157 binary digits. n = 736: after 100000 steps, the iterate has 135 binary digits. n = 736: after 150000 steps, the iterate has 131 binary digits. n = 736: after 200000 steps, the iterate has 507 binary digits. n = 736: after 250000 steps, the iterate has 414 binary digits. java.lang.StringIndexOutOfBoundsException: Range [57, 1) out of bounds for lengt n = 736: after 350000 steps, the iterate has 465 binary digits. n = 736: after 400000 steps, the iterate gap> positions2 := List(Cycle(h,33),n->Posit,); java.lang.StringIndexOutOfBoundsException: Range [21, 1) out of bounds for lengt n = 896: [[0 ]] n = 896: after 100000 steps]]> , ,[ 2,2 ,[ 24,[ ,2 ] 26, 2] 32 6296, [ [86,2 , ,8] 106 ,[112,104], [ , , [ 122, 2 ], [ 136, 440 ], [ 142, 2 ], [ pretty <></>java.lang.StringIndexOutOfBoundsException: Ind [ 166, 2 ], [ 172, 8 ], [ 176, 24 ], [ 182, 2 ], [ 196, 8 ], [ 202, 2 ], [ 206, 2 ], [ 212, 8 ], [ 226, 2 ], [ 232, 24 ], [ 236, 8 ], [ 242, 2 ], [ 256, 56 ], [ 262, 2 ], [ 266, 2 ], [ 272, 408 ], [ 286, 2 ], [ 292, 8 ], [ 296, 104 ], [ 302, 2 ], 316 ] [322 ,[, 2] 332,8 ] [346, , [ 352, 264 ], [ 356, 8 ], [ 362, 2 ], [ 376, 1304 ], [ 382, 2 ], [ 386, 4, 47 ,51,67,6223,6221, 69 6215, 6205,6209,6211 4,2] 436 ,[442,2 ] 446, ] [ 452, ,java.lang.StringIndexOutOfBoundsException: Index 62 o [ 466, 2 ], [ 472, 104 ], [ 476, 8 ], [ 482, 2 ], [ 496, 24 ], [ 502, 2 ], [ 506, 2 ], [ 512, 696 ], [ 526, 2 ], [ 532, 8 ], [ 536, 3912 ], [ 542, 2 ], [ 556, 8 ], [ 562, 2 ], [ 566, 2 ], [ 572, 8 ], [ 586, 2 ], [ 592, 888 ], [ 596, 8 ], [ 602, 2 ], java.lang.StringIndexOutOfBoundsException: Index 67 out of bounds for length 67 [ 652, 8 ], [ 656, 24 ], [ 662, 2 ], [ 676, 8 ], [ 682, 2 ], [ 686, 2 ], [ 692, 8 ], [ 706, 2 ], [ 712, 24 ], [ 716, 8 ], [ 722, 2 ], [ 736, 495448 ], [ 742, 2 ], [ 746, 2 ], [ 752, 1272 ], [, 2],[772,8 ] , ,[782 2] 796 ,java.lang.StringIndexOutOfBoundsException: Index 64 ou [ 802, 2 ], [ 806, 2 ], [ 812, 8 ], [ 826, 2 ], [ 832, 120 ], [ 836, 8 ], [ 842, 2 ], [ 856, 2264 ], [ 862, 2 ], [ 866, 2 ], [ 872, 24 ], [ 886, 2 ], [ 892, 8 ], [ 896, 132760 ], [ 902, 2 ], [ 916, 8 ], [ 922, 2 ], [ 926, 2 ], [ 932, 8 ], [ 946, 2 ], [ 952, 456 ], [ 956, 8 ], [ 962, 2 ], [ 976, 24 ], [ 982, 2 ], [ 986, 2 ], [ 992, 1064 ] ] ]]> </Example> So far the author has checked that all positive integers less than 173176 lie in finite cycles of <M>g</M>. Several of them are longer than 1000000, and the cycle containing 25952 has length 245719352. Whether the cycle containing 173176 is finite or infinite has not been checked so far -- in any case it is longer than 5700000000, and it exceeds <M>10^{40000}</M>. Presumably it is finite as well, but checking this may require a lot of computing time. <P/> On the one hand the cycles of <M>g</M> seem to behave <Q>randomly</Q>, perhaps as if they would ascend or descend from one point to the next by a certain factor depending on which side a thrown coin falls on. -- In this <Q>model</Q>, cycles would be finite with probability 1 since the simple random walk on &ZZ; is recurrent. On the other, there seems to be quite some structure on them, however little is known so far. <P/> First we observe that each orbit under the action of <M>G</M> seems to split into two cycles of <M>h := abcacb</M> of the same length (of course this has been checked for many more orbits than those shown here): <Example> <![CDATA[ gap> h := a*b*c*a*c*b; <rcwa permutation of Z with modulus 360> gap> List(CyclesOnFiniteOrbit(G,h,32),Length); [ 3148, 3148 ] gap> List(CyclesOnFiniteOrbit(G,h,736),Length); [ 247724, 247724 ] ]]> </Example> One cycle seems to contain the points at the odd positions and the other seems to contain the points at the even positions in the cycle of <M>g</M>: <Example> <![CDATA[ gap> cycle_g := Cycle(g,32);; gap> positions1 := List(Cycle(h,32),n->Position(cycle_g,n));; gap> Collected(positions1 mod 2); [ [ 1, 3148 ] ] gap> positions2 := List(Cycle(h,33),n->Position(cycle_g,n));; gap> Collected(positions2 mod 2); [ [ 0, 3148 ] ] ]]> </Example> However the ordering in which these points are traversed looks pretty <Q>scrambled</Q>: <Example> <![CDATA[ gap> positions1{[1..200]}; [ 1, 6271, 6291, 6281, 6285, 6287, 6283, 6289, 6273, 6275, 6277, 6279, 6293, 5, 15, 17, 19, 6259, 6261, 6263, 6265, 21, 23, 25, 41, 6227, 6229, 6231, 6233, 6235, 6237, 6239, 43, 53, 55, 57, 63, 59, 61, 65, 45, 47, 49, 51, 67, 6223, 6221, 69, 6163, 6215, 6205, 6209, 6211, 6207, 6213, 6165, 6171, 6177, 6179, 6181, 6183, 6175, 6173, 6185, 6189, 6191, 6187, 6193, 6169, 6167, 6195, 6199, 6201, 6197, 6203, 6217, 73, 83, 85, 87, 103, 113, 115, 117, 4357, 4361, 4363, 4359, 4365, 4371, 4373, 4375, 4377, 4369, 4367, 4379, 119, 121, 123, 125, 129, 131, 127, 133, 139, 141, 143, 145, 137, 135, 147, 149, 151, 153, 155, 159, 161, 157, 163, 169, 175, 4283, 4281, 177, 4271, 4273, 4275, 4277, 181, 4255, 4257, 4259, 261, 4263, ,,2161,2163, 4195, 4199, 4201, 4197, 4203, 4209, 4211, 4213, 4215, 4207, 4205, 4217, 2165, 2167, 2169, 2171, 2175, 2177, 2173, 2179, 2185, 2187, 2189, 2191, 2183, 2181, 2193, 2195, 2197, 2199, 2201, 2467, 2469, 4117, 4121, 4123, 4119, 4125, 4131, 4133, 4135, 4137, 4129, 4127, 4139, 2471, 2473, 2475, 2477, 2487, 2489, 2491, 2507, 2517, 2519, 2521, 2537, 3923, 3925, <[DATA[ ]]> </Example> </Section> <!-- #################################################################### --> <="FourTransitiveGroup> <Heading> A group which acts 4-transitively on the positivegap>tups:= tups,up>ForAll[1,]31,2,4,[,2]j </Heading> In this section, we would like to show that the group <M>G</M> generated by the two permutations <java.lang.StringIndexOutOfBoundsException: Index 4 out of bounds for length 4 <![CDATA[ gap> a := RcwaMapping(java.lang.StringIndexOutOfBoundsException: Index 0 out of boun gap> u := RcwaMapping([[3,0,5],[9,1,5],[3,-1,5],[9,-2,5],[9,<!CDATAjava.lang.StringIn gap> SetName(a,"a"); 256, 400,,400 100 32,324,400 ,400 ,25 35, ]]> </Example> which we have already investigated in earlier examples acts 4-transitively on the integersjava.lang.StringIndexOutOfBoundsException: Index 32 out of bounds f Obviously1875,75 225, 405, 225, 225,675 ,2187, 729, 2187,216 , First we show that 243, 2187, 1944, 2187 ,,576 ,,81 729, 2187 We start by checking in which residue classes sufficiently large positive integers are mapped to smaller ones by a suitable group element: <Example> <![CDATA[ gap> List([a,a^-1,u,u^-1],DecreasingOn); [ 1(2), 0(3), 0(5) U 2(5), 2(3) ] gap> Union(last); Z \ 4(30) U 16(30) U 28(30) ]]> <Example> We see that we cannot always choose such a group element from the set of generators and their inverses -- otherwise the coefficient of the respective mapping are mapp <C>Integers</C>. <[DATA[ java.lang.StringIndexOutOfBoundsException: Range [10, 9) out of bounds for lengt gap> List([a,a^-1,u,u^-1,a^2,a^-[ ,8 25 9,9 16,100 , 50,, 75 ,81 ,81 ,400, [1(), () 5)U 25, 2(),(8 U () 03 U2()U (), 0(25) U 12(25) U 17(25) U 20(25), 2(3) U 1(9) U 3(9) ] gap> Union(last); # Still not enough ... Z \ 4(90) U 58(90) U 76(90) java.lang.StringIndexOutOfBoundsException: Range [21, 3) out of bounds for lengt > DecreasingOn); [ 1(2), 0(3), 0(5) U 2(5), 2(3), 1(8) U 7(8), 0(3) U 2(9) U 7(9), 0(25) U 12144 324 972,400, ,80 80 ,2500,100 1250,625,625,25, 3(5) U 0(1045 135 600 120,150,1875 ,225,135,225,225,675 gap> Union(last); # ... but that's it! Integers ]]> </Example> Finally, we have to deal with <Q>small</Q> integers. java.lang.StringIndexOutOfBoundsExcepti the coefficients of rcwa mappings introduced Since the least common multiple of the moduli of t Let <M>c_{r(m)} > a_{r(m)}</M>. Then we easily see that <M>(a_{r(m)}n+b_{r(m)})/c_{r(m)} > n</M> implies <M &;b_{r(m}/c_rm}a_rm))<M. Thus we can restrict our considerations to integers <Mn&;b_{rm }/> <>{\m max<M>isthelargest entry acoefficienttripleofone of the group elements in our list: <ExampleExample> <![CDATA[ gap> List([a,a^-1,u,u^-1,a^2,a^-2,u^2,u^-2,a > f->Maximum(List(Coefficients(f),c->c[2]))); [ 1, 1, 4, 2, 7, 7, 56, 28, 25, 17, 17, 11unionof30 classes(od gap> Maximum(last); 56 ]]> </Example> Thus this upper bound is 56. The rest is easy -- all we have to do is to check that the orbit We have to treat 4 separately integers less than or equal to 56: <Example> <[DATA[ gap> S := [1];; gap> while not IsSubset(S,[1..56]) do > S := Union(S,S^a,S^u,java.lang.StringIndexOutOfBoundsException: Index 1 out of bou > od; gap> IsSubset(S,[1..56]); true ]]> </Example> Checking 2-transitivity is computationally harder, and in the sequel we will omit some steps which are in practice needed to find out <Q>what to do</Q>. The approach taken here is to show that the stabilizer of 1 in <M>G</M> acts transitively on the set of positive integers greater than 1. We do this by similar means as used above for showing the transitivity of the action of <M>G</M> on the positive integers. We start by determining all products of at most 5 generators and their inverses, which =[,]java.lang.StringIndexOutOfBoundsException: Index 18 out of bou not suffice!): <Example> <![CDATA[ gap> gens := [a,u,a^-1,u^-1];; gap> tups := > stabelm := List,(t,i>gens[]);; gap> gap> decs := List(stabelm,DecreasingOn);; 1364 gap> tups := Filtered(tups,tup->ForAll([[1,3],[3,1],[2,4],[4,2]], > l->PositionSublist(tup,l)=fail));; gap> Length(tups); 484 gap> stab> S: ; gap> for tup in tups do > n := 1; >p don =n[] > if n = 1 then Add(stab ifS< S_oldthenViewObj(S)Print("); fijava.lang.StringIndexOutOfBoun > od; gap> Length(stab); 118 gap> stabelm := List(stab,tup->Product(List(tup,i->gens[i])));; gap> ForAll(stabelm,elm->1^elm=1); # Check. true ]]> <Example> notquitesmallmoduli: <Example> <![CDATA[ gap> List(stabelm,Modulus); [ 4, 3, 16, 25, 9, 81, 64, 100, 108, 100, 25, 75, 27, 243, 324, 243, 256, 400, 144, 400, 100, 432, 324, 400, 80, 400, 625, 25, 75, 135, 150, 75, 225, 81, 729, 486, 729, 144, 144, 81, 729, 1296, 729, 6561, 1024, 1600, 192, 1600, 400, 576, 432, 1600, 320, 1600, 2500, 100, 100, 180, 192, 192, 108, 972, 1728, 972, 8748, 1600, 400, 320, 80, 1600, 500300 2500 ,,75,675 ,75135 ,600 ,600 1875, 75, 225, 405, 225, 225u residueclasses(od 5760)(classes> 243, 2187, 1944, 2187, 19683, 576, 144, 576, 432, 81, 81, 729, 2187, 5184, 324, 8748, 243, 2187, 19683, 26244, 19683 ] gap> Lcm(last); 12597120000 gap> Collected(Factors(last)); [ [ 2, 10 ], [ 3, 9 ], [ 5, 4 ] ] ]]> </Example> Similar as before, we determine for any of the above mappings the residue classes whose elements larger than the java.lang.StringIndexOutOfBoundsExceptio - coefficient of the respective mapping are mapped to smaller integers: <Example> <!union of32 residue (od > gap> decs := List(stabelmes (mod ) s); [ 2, 3, 8, 25, 9, 9unionof20residueclasses mod259200) (16> 48, 200, 100, 72, 108, 400, 80, 200, 625, 25, 75, 45, 75, 75, 225, 81, ,81 ,144, 144 ,243,216, 243 , ,64,400,java.lang.StringIndexOutOfBoundsException: Index ,48 144,1600,320 400,2500 100 ,60,96, ,108 , 144, 324, 972, 400, 400, 80, 80unionof 12 residue classes m 259200 (10classes) 75, 75, 75, 45, 135, 600, 120, 150, 1875, 75, 225, 135, 225, 225, 675, 243, 729, 243, 729, 108, 216, 243, 729, 162, 729, 2187, 144, 144, 144, 144, 81, 81, 243, 729, 1296, 324, 972, 243, 729, 2187, 1458, 2187 ] gap> Lcm(last); ]]> </Example> Since the least common multiple of the moduli classes is as large as 174960000, directly forming their union and ckingwhether isequal tothe setof takerelatively much and . starting java.lang.StringIndexOutOfBoundsException: Range [45, 44) out of subtracting the above sets one-by-one in a suitably chosen order is cheap: <Example> <![CDATA[ gap> SortParallel(decs,stabelm, > function(S1,S2) > return First([1..100g> :=(tup>[13,31,24,4,2]java.lang.StringIndexOutOfBoundsExceptio > < First([1..100],k->Factorial(k) mod Modulus(S2)=0); > end);>ap> for in tups do gap : Integers; fori in [.Length()]do > S_old := S; S := Difference(S,> if 1,, Add(,tup;fi; > if S <> S_old then ViewObj(S); Print("\n"); fi; > if S = [] then maxind := i; break; fi; > od; 0(2) 2(6) U 4(6) <union of 19 residue classes (mod 60) (6 classes)> <union of 8 residue classes (mod 30)> <union of 120 residue classes (mod 720)> <union of 112 residue classes (mod 720)> <union of 80 residue classes (mod 720)> <union of 24 residue classes (mod 720)> <union of 16 residue classes (mod 720) (12 classes)> <union of 8 residue classes (mod 720)> <union of 6 residue classes (mod 720)> 4(720) U 94(720) U 148(720) U 238(720) <of classesm ) <union of 72 residue classes (mod 51840 functionS1) <java.lang.StringIndexOutOfBoundsException: Range [7, 6) out of bounds for length union of 04 classes mod259200> <union of 144 residue classes (mod 259200)> <union gap>S : Integers; <union of 84 residue classes (mod > <union of 48 residue classes (mod 259200)> <union of 24 residue classes (mod 259200)> <union of 12 residue classes (mod 259200) (6 classes)> <union of 30 residue classes (mod 777600)> 54004(64800) U 151204(259200) U 216004(259200) [ ] ]]> <> Similar as above, it remains to check that the <Q>small</Q> integers all lie in the orbit containing 2. Obviously, it is sufficient to check that any integer greater than 2 is mapped to a smaller one by some suitably chosen element of the stabilizer under consideration: <Example> <![CDATA[ gap> Maximum(List(stabelm{[1..maxind]}, > f->Maximum(List(Coefficients(f),c->c[2])))); 6581 gap> Filtered([3..6581],n->Minimum(List(stabelm,elm->n^elm))>=n); [ 4 ] ]]> </Example> We: <Example> <![CDATA[ 1(u**^2*^1u; 1 gap> 4^(u*a*u^2*ajava.lang.StringIndexOutOfBoundsException: Index 43 out of bounds 3 ]]> </Example> java.lang.StringIndexOutOfBoundsException: Index 74 out of bounds for length 74 u of12 residue classes m 129600> as 2, thus that this stabilizer acts transitively on <M>&NN; \setminus \{1\}</M>. But this means that we have established the 2-transitivity of the action of <M>G</M> on &NN;. <P/> nthe following, repeatthe above steps to show that this action is indeed 3-transitive: <Example> <![CDATA[ gap> tups := Concatenation(List([1..6],k->Tuples([1..4],k)));; gap> tups := Filtered(tups,tup->ForAll([[1,3]union 22 residueclasses(> > >PositionSublisttup,)fail); gap> stab := [];; gap> for tup in tups do > l := [1,2]; > for i in tup do l := List([ ] > if l = [1,2] then Add(stab,tup); fi; > od; Lengthstab)java.lang.StringIndexOutOfBoundsException: Index 18 out of bounds for 212 gap> stabelm gap>smallnum : [..]; gap> decs := List(stabelm,DecreasingOn);; ap SortParallel(,function(S1S2 > return First([1..100],k->Factorial(k) mod Mod(S1)=0) > < First([1..100],k->Factorial(k) mod Mod(S2)=0); end); gap> S := Integers;; gap> for i in [1..Length(decs)] do > S_old := S; S := Difference(S,decs[i]); > if S <> S_old then ViewObjjava.lang.StringIndexOutOfBoundsException: Index 33 out of bo > if S = [] then break; fi; > od; Z \ 1(8) U 7(8) <union of 424 residue classes (mod 720)> <union of 169 residue classes (mod 720)> <union of 51 residue classes (mod 720)> <union of 45 residue classes (mod 720)> <union of 42 residue classes (mod 720)> <union of 35 residue classes (mod 720)> < 30 classes m )java.lang.StringIndexOutOfBoundsException: Range [39, 40) out of boun <union of 16 residue classes (mod 720) (10 classes)> <union of 11 residue classes (mod 720) (7 classes)> <union of 8 residue classes (mod 720) (6 classes)> 148(360) U 238(360) U 4(720) U 454(720) 238(360) U 4(720) U 148(720) U 454(720) <nionof 28residue ( 5760> <union of 24 residue classes (mod 5760)> < of23 residue classes (mod 5760)> <union of 22 residue classes (mod 5760)> <union of 20 residue classes (mod 5760) (14 classes)> <union of 19 residue classes (mod 5760) (14 classes)> <union of 16 residue classes (mod 5760) (12 classes)> <union of 112 residue classes (mod 51840)> <union of 96 residue classes (mod 51840)> <union of 90 residue classes (mod 51840)> <union of 51 residue classes (mod 51840)> <union of 21 residue classes (mod 51840)> <union of 19 residue classes (mod 51840) (15 classes)> <union of 16 residue classes (mod 51840) (12 classes)> <union of 54 residue classes (mod 259200)> <union of 53 residue classes (mod 259200)> <union of 50 residue classes (mod 259200)> <union of 38 java.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for le <union of 35 residue classes (mod 259200)> <union of 32 residue classes (mod 259200)> <union of 24 residue classes (mod 259200)> <union of 22 residue classes (mod 259200)> <union of 20 residue classes (mod 259200) Heading> <union of 18 residue classes (mod 259200) (15 classes)> <of residueclasses (mod 259200) (13 classes)> <union of 15 residue classes (mod 259200) (12 classes)> <java.lang.StringIndexOutOfBoundsException: Range [7, 6) out of bounds for length <union of 7 residue classes (mod 259200)> 2164(259200) U 66964(259200) U 228964(259200) [ ] gap> Maximum(List(stabelm,f->Maximum(List(Coefficients(f),c->c[2])))); 515816 gap> smallnum := [4..515816];; gap> iin 1.() > smallnum := Filtered(smallnum,n->n^stabelmgap> (G) > od; gapgap (.^2G.2^*(12G2^;#G >the product C_infty C_2. [ ] ]]> </Example> Thesame for4-transitivity <Example> <![CDATA[ gap> tups := Concatenation(List([1..8],k->Tuples([1..4],k)));; :Filteredtup-ForAll([,],[1[,][,], > l->PositionSublist(tup,l)=fail));; gap> stab := [];; java.lang.StringIndexOutOfBoundsException: Index 26 out of bounds for length 23 > =[,,]java.lang.StringIndexOutOfBoundsException: Index 20 out of bounds for lengt >| nifnin (4java.lang.StringIndexOutOfBoundsException: Index 30 out of bounds for l > if l = [1,2,3] then Add(stab,tup); fi; > od; gap> Length(stab); 528 gap> stabelm := [];; gap> for i in [1..Length(stab)] do > elm := One(G); > jin i java.lang.StringIndexOutOfBoundsException: Index 26 out of bounds for length > if Modulus(elm) > 10000 then elm := fail; break; fi; > elm := elm * gens[j]; > od; > if elm <> fail then Add(gap> gens := [ClassShift(0,1)^-1,Class12,,), > od; gap> Length(stabelm); 334 gap> decs := List(stabelm,DecreasingOn);; gap> SortParallel(decs,stabelm, > function(S1,S2) > return First([1..100],k->Factorial(k) mod Modulus(S1) = 0) > < First([1..100],k->Factorial(k) java.lang.StringIndexOutOfBoundsException: Index 3 > end); gap> S := Integers;; gap> for i in [1..Length(decs)] do > S_old := S; S := Difference(S,decs[i]); > if S <> S_old then ViewObj(S); Print("\n"); fi; > if S = [] then maxind := i; break; fi; > od; Z \18 U () <union of 24 [, ],[ , , 16,3 <union of 22 gap>decs :ListstabelmDecreasingOn; <[ 04) 34, 0(),3) 24 () () () 2() (4,1(, 4(18) <union of 28 residue classes (mod 576)> <union of 26 residue classes (mod 576)> <union of 21 residue classes (mod 576)> <union of 20 residue classes (mod 576) (7 classes)> <union of 18 residue classes (mod 576) (8 classes)> <Similar theprevious section remains tocheck thatthe integers <union of 15 residue classes (mod 576) (6 classeswith Qsmall<Q> absolute liein the containing&;1 un <union of 120 residue classes&bsp;0: <union of 45 residue classes (mod 5184)> <union of 30 residue classes (mod 5184)> <union of 28 residue classes (mod 5184)> <union of 6 <union of 6 residue < 116residue classes ( )> <union of 104gap>for doS : Union(S,^elmS(elm^1)) ; <union of 92 residue classes (mod 32400)> <union of 84 residue classes (mod 32400)> <union of 80 residue classes (mod 32400)> <union of 210 residue classes (mod 129600)> <nionof189 residue classes (mod 129600> <union of 160 residue java.lang.StringIndexOutOfBoundsException: Index 4 out of bound <union of 136 residue classes (mod 129600)> <union of 133 residue classes (mod 129600)> <union of 122 residue classes (mod 129600)> <union of 114 residue classes (mod 129600)> <union of 106 residue classes (mod 129600)> <of 104 classes mod 129600> <union of 100 residue classes (mod 129600)> <)> <union of 60 residue classes (modelements areq , looking at setsof is cheapjava.lang.StringIn <union of 52 residue classes (mod 129600)> <union of 48 residue classes (mod 129600)> <union of 40 residue classes (mod 129600)> <union of 36 residue classes (mod 129600)> < classes mod129600) <union of 24 residue classes (mod 129600)> <union of 16 residue classes (mod 129600) (10 classes)> <union of 12 residue classes (mod 129600)> <union of 10 residue classes (mod 129600)> <union of 8 residue classes (mod 129600)> <union of 6 residue classes (mod 129600)> 57406(129600) U 63076(129600) U 115006(129600) U 120676(129600) 57406(129600) U 115006(129600) U 192676(259200) U 250276(259200) <union of 15 residue classes (mod 777600) (6 classes)> <union of 9 residue classesgap Lengths; <union of 30 residue classes (mod 3110400)> <union of 26 residue classes (mod 3110400)> <union of 22 residue classes (mod 3110400)> <union of 19 residue classes (mod 3110400) (10 classes)> <union of 14 residue classes (mod 3110400) (8 classes)> 705406(777600) U 2007076(3110400) U 2649406(3110400) U 2784676(3110400) <union of 14 residue classes (mod 9331200) (8 classes)> 1483006(2332800) gap>decsp: List(tabelmelm-(9.]n>^<) <union[[9 , ,12 ,16 ] ,16 ] [9 13 ] [12 ] 2784676(9331200) U 5117476(9331200) [ ] [9 11 13 15]] > >Uniond)java.lang.StringIndexOutOfBoundsException: Index 18 out of bounds for len 37387 gap> smallnum := [5..37387];; gap> i 1.] java.lang.StringIndexOutOfBoundsException: Range [28, 29) out of bounds f > smallnum := Filtered(smallnum,n->n^stabelm[i]>=n); > od; gap> smallnum; [ ] ]]> </Example> There is even some evidence that the degree of transitivity of the action of <M>G</M> on the positive integers is higher than It checkthat group<>G/>doesnotact4-ransitively <Example> <![CDATA[ gap> phi := EpimorphismFromFreeGroup(G); [ a, u ] -> [ a, u ] gap> F := Source(phi); f groupon generators [ a, u ]> gap> List([5..20], > n->RepresentativeActionPreImage(G,[1,2,3,4,5], > [1,2,3,4,n],OnTuples,F)); [ <identity ...>, a^-3*u^4*a*u^-2*a^2, a^-1*(a^-1*u)^4*a^-1*u^-1*a, a^4*u^-2*a^-4, a^-1*u^-4*a, (u^2 od; a^-1*u^2*a, a^-1*u^-6*a, a^2*u^4*a^2*u^2, u^-4*a*u^-2*a> end;java.lang.StringIndexOut a^-1*u^->old: (orb; u^-2*a*u^2*a*u^-2, u^-4*a^2*u^2 ] ]]> </Example> Enter <C>AssignGlobals(LoadRCWAExamples().CollatzlikePerms);</C> in order to assign the global variables defined in this section. </Section> <!-- #################################################################### --> <Section Label="sec:ThreeButNotFourTransitiveGroup"> <Heading> A group which acts 3-transitively, but not 4-transitively onjava.lang.StringIndexOutOfBo </Heading> In this section, we would like to show that the group <M>G</M> generated by acts This showsthat MG</>actsnot4-transitivelyon &ZZ; <Example> <![CDATA[ gap> G := Group(ClassShift(0,1),ClassTransposition(1,2,0,4)); <rcwa group over Z with 2 generators> gap> IsTame(G); false gap> (G.1^-2*G.2)^3*(G.1^2*G.2)^3; # G <> the free product C_infty * C_2. IdentityMapping( Integers ) gap> Display(G:CycleNotation:=false); Wild rcwa group over Z, generated by [ Tame rcwa permutation of Z: n -> n + 1 Rcwa java.lang.StringIndexOutOfBoundsException: Index 9 out of bounds for length 9 | 2n-2 if n in 1(2) n |-> < (n+2)/2 if n in 0(4) | n if n in 2(4) \ ] ]]> </Example> This /Section group generated by the first of the two java.lang.StringIndexOutOfBoundsException: Index show that it acts 2-transitively. We essentially proceed as in the example in the previous section, by checking that the stabilizer of 0 acts transitively on <M>&ZZ; \java.lang.StringIndexOutOfBoundsException: Range [0, 38) out java.lang.StringIndexOutOfBoundsException: Index 67 out of bounds for length 9 <![CDATA[ gap> gens := [ClassShift(0,1)^-1,ClassTransposition(1,2,0,4), > ClassShift(0,1)];; gap> tups := Concatenation(List([1!CDATAjava.lang.StringIndexOutOfBoundsException: R gap> tups := Filtered(tups,tup->ForAll([[0,0],[-1,1],[1,-1]], > l->PositionSublist(tup,l)=fail));; gap> Length(tups); java.lang.StringIndexOutOfBoundsException: Index 28 out of bounds for length 3 gap> stab := [];; gap> for tup in tups do > n := 0; > for i in tup do n := n^gens[i+2]; od; > if n = 0 then Add(stab,tup); fi; > od; gap> =Liststabtup-Product(ist(,i>ensi2)); gap> Collected(List(stabelm,Modulus)); [ [ 4, 6 ], [ 8, 4 ], [ 16, 3 ] ] gap> decs := List(stabelm,DecreasingOn); [ 0(4), 3gap> ((1.1000]n-Trajectory(,n,S0)))java.lang.StringIndexOutOfBoundsExceptio 0(8), 3(8) ] gap> Union(decs); Integers ]]> </Example> Similar as in the previous section, it remains to check that the integers with <Q>small</Q> absolute value all lie in the orbit containing 1 under the actionofthe stabilizer ofn;0 <Example> <![CDATA[ gap> Maximum(List(stabelm,f->Maximum(List(Coefficients(f), > c- 21 gap> S := [1];; gap> for elm in stabelm do S := Union(S,S^elm,S^(elm^-1)); od; gap> IsSubset(S,Difference([-21..21],[0])); # Not yet .. false gap> for elm in stabelm do S := Union(S,n |- 113/ () gap> IsSubset(S,Difference([-21..21],[0])); # ... but now! true ]]> </Example> Now we have to check for 3-transitivity. Since we cannot find for java.lang.StringIndexOutOf residue class an element of the pointwise stabilizer of <M>\{0,1\}</M> which properly divides its elements, we also have to take additions and subtractions into consideration. Since the moduli of all of our stabilizer elements are quite small, java.lang.StringIndexOutOfBoundsException: Index 29 out of <Example> <![CDATA[ gap> tups := Concatenation(List([1..10],k->Tuples([-1,0,1],k)));; gap>tups := Filtered(tups,tup-ForAll([00,-,],[-], > l->PositionSublist(tup,l)=fail));; gap> Length(tups); 3069 stab:=[]; gap> for tup in tups do > l := [0,1]; > for i in tup do l := List(l,n->n^gens[i+2]); od; > if l = [0,19,6,23515] > od; gap> Length(stab); 10 gap> stabelm := List(stab,tup->Product(List(tup,i->gens[i+2])));; gap> Maximum(List(stabelm,Modulus)); 8 gap> Maximum(List(stabelm, > f->Maximum(List(Coefficients(f),c->AbsInt(c[2]))))); 8 gap> decsp := List(stabelm,elm->Filtered([9..16],n->n^elm<n)); 9 13] ,12 14 16 ] [12 ,[9 13 ,[ ,16 ] [ 9, 11, 13java.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for len [ 9, 11, 13, 15 ] ] gap> Union(decsp); [ 9 .. 16 ] gap> decsm := List(stabelm,elm->Filtered([-16..-9],n->n^elm>n thejava.lang.StringIndexOu 15 13 -,- ,[-6 - ] [ 16,-12],[ 15 11 , [ -16, -14, -12, -10 ], [ -15, -11 ], [ -15, -11 ], [ -16, -14, -12, -10 ], [ -16, -14, -12, -10 ], [ -15, -11 ] ] gap> Union(decsm); [ -16 .. -9 ] gap> S := [2]Enter <AssignGlobalsL).SlowlyContractingMappings;</> gap> for elm in stabelm do S := Union(S,S^elm,S^(java.lang.StringIndexOutOfBoundsExceptio gap> IsSubset(S,Difference([-8..8],[0,1])); true ]]> </Example> At this point 3-ransitivity. java.lang.StringIndexOutOfBoundsException: Index 1 out of bounds for length 0 We do this by checking that it is not transitive on 4-tuples (mod 4). Since <M>n</M> < =sec:" agenerator MG/M> m , it tocompute(od : <Example> <![CDATA[ gap> orb := [[0,1,2,3]];; gap> extend := function () java.lang.StringIndexOutOfBoundsException: Index 263 out of bounds for length 17 > for gen in gens do > =UnionorbList(,List(,-n) mod8)); > java.lang.StringIndexOutOfBoundsException: Range [10, 11) out of bounds for leng >; >old :ShallowCopy(rb; > extend(); Print(Length(orb),"\n"); > until orb = old; 7 27 97gapjava.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 279 573 916 1185 1313 1341 java.lang.StringIndexOutOfBoundsException: Range [4, 5) out of bounds for length 1344 S(Listorbl>l mod4)); p<id""name="X7950ED88879EE36F<a<p> gapp the main ofthe to Userwithtools S Union(,S^T) true ] </Example> This shows that <M>G</M> acts not 4-transitively on &ZZ;. corresponding calculationfor 3nionof810 classes mod> <> <![CDATA[ gap> orb := [[0,1,2]];; gap> repeat > old := ShallowCopy(orb); > extend(); Print(Length(orb),"\n"); > until 7 27 84 207 363 459 503 512 512 gap> Length(Set(List(orb,l->l mod 4))); 64 gap> last = 4^3; true ]]> </Example> Needless to say that the latter kind of argumentation is java.lang.StringIndexOutOfBoundsE for proving, but only for disproving <M>k</M>-transitivity. </Section> <!-- #################################################################### --> <Section Label="sec:SlowlyContractingMapping"> <Heading> An rcwa mapping which seems gap> x := Indeterminate(GF(4),1);; SetName(x,"x"); </Heading> The iterates of an integer under the Collatz mapping <M>T</M> seem to approach its contraction centre -- this is the finite set where all trajectories end up after a finite number of steps -- rather quickly and do not get very large before doing so (of course this is a purely heuristic statement as the <M>3n+1</M> conjecture has not been <Example> <![CDATA[ gap> T := RcwaMapping([[1,0,2],[3,1,2]]);; gap> S0 := LikelyContractionCentre(T,100,1000); #I Warning: `LikelyContractionCentre' is highly probabilistic. The returned result can only Rcwa mappingofGF(2[ modulus x See ?LikelyContractionCentre for more information. [136,-,-82,-68,-, -, 41 37 -, 25,-, 10 -,-5,java.lang.StringIndexOutOfBoundsException: - ,1 ] true gap> [ 1gap> List[,L2]IsSurjective 11, 11, 8, [true,true java.lang.StringIndexOutOfBoundsException: Index 14 out of bou gap> Maximum(List([1..1000],n->Length(Trajectory(T,n,S0)))); 113 gap> Maximum(List([1..1000],n->Maximum(Trajectory(T,n,S0)))); 125252 ]]> </Example> msto be contracting well,but are much longer: <Log> <![CDATA[ gap> f6 := RcwaMapping([[ 1,0,6],[ 5, 1,6],[ 7,-2,6], > [11,3,6],[11,-2,6],[11,-1,6]]);; gap> Display(f6); Rcwa mapping of Z with modulus 6 / | n/6 if n in 0(6) | (5n+1)/6 if n in 1(6) | (7n-2)/6 if n in 2(6) n |-> < (11n+3)/6 if n in 3(6) | (11n-2)/6 if n in 4(6) | (11n-1)/6 if n in 5(6) | \ gap> S0 := LikelyContractionCentre thetwo states 0&;nbsp;<><M I : LikelyContractionCentre is java.lang.StringIndexOutOfBoundsException: Range [49, The returned result can only be regarded as a See ?LikelyContractionCentre for more information. Trajectory(,25,) [ 25, 21, 39, 72, 12, 2 ] gap> List([1..100],n->Length(Trajectory(f6,n,S0))); [ 1, 1, 3, 4, 1, 2, 3, 2, 1, 5, 7, 2, 8, 17, 3, [1 ,0 ,0java.lang.StringIndexOutOfBoundsException: 5, 5, 6, 1, 4, 2, 15, 1, 1, 3, 2, 5, 13, 3, 2, 3, 4, 1, 8, 4, 4, 2, 7, 19, 23517, 3, 9, 3, 1, 18, 14, 2, 20, 23512, 14, 2, 6, 6, 1, 4, 19, ,23511,8, 23513, 10, 1,13,13 ,1 23517 ,, 9 ,6 12, 8, 6, 18, 14, 23516, 31, 12, 23545, 4, 21, 19,1 1 1 1 ,1 ,,1,, 9 6 ] gap> Maximum(Trajectory(f6,47,S0)); 7363391777762473304431877054771075818733690108051469808715809256737742295\ 45698886054 ]]> </Log> Computing the trajectory of 3224 takes quite a while -- this trajectory ascends to about <M>3 \cdot 10^{2197}</M>, before it approaches the fixed point 2 after 19949562 steps. <P/> henconstructing C< of java.lang.StringIndexOutOfBoundsException: Index 64 out of bou partial mappings have been chosen to been chosen to be numbers coprime to the common denominator, whose product is just a little bit smaller than the <C>Modulus(f6)</C>th power of the denominator. In the example we have <M>5 \cdot 7 \cdot 11^3 = 46585</M> and <M>6^6 = 46656</M>. <P/> Although the trajectories of <C>T</C> are much shorter than those of <<C> seemsthatthisdoes java.lang.StringIndexOutOfBoundsException: Range [55, 54) out whether the mapping <C>T</C> is contracting essentially easier -- java.lang.StringIndexOutOfBoundsException: Range [18, 17) out of bounds for leng the cases ..forexamplewhenthere some<Mk/M>suchthat applying the <M>k</M>th power of the respective mapping to java.lang.StringIndexOutOfBoundsEx decreases its absolute valueclass"APprompt"gap;/span < class"> Enter "example"< in order to java.lang.StringIndexOutOfBoundsException: Range [0, 18) out of bounds </Section> - ########################################- <java.lang.StringIndexOutOfBoundsException: Range [9, 8) out of bounds for length <Heading>Checking a1 3 5 67java.lang.StringIndexOutOfBoundsException: Index 78 out of InCite Key"Andaloro00/,P ndaloro has that java.lang.StringIndexOutOfBoundsException: trajectories of integers <M>n \in 1(16)</M> under the Collatz mapping always java.lang.StringIndexOutOfBoundsException: Range [8, 7) out of bounds for length /pre</div> the imagesof theresidueclass ()under powersoftheCollatz java.lang.StringIndexOutOfBoundsException: Range [13, 7) out of bounds for lengt .Thusweput <>S: 116/> successively unite the et&;M></> java.lang.StringIndexOutOfBoundsException: Range [23, 22) out of bounds for < <![CDATA[ T =RcwaMapping([1,0,],312]; <o Z modulus2> S:=ResidueClass(ntegers16) 1(<<p>Computetheof spanclass="impleMath"><span> in <span class="SimpleMath">G</span>, i.e. compu dl 1(16) U 2(24) gap> S := Union(S,S^T); 1(12) U 2(24) U 17(48) U 33(48) java.lang.StringIndexOutOfBoundsException: Range [4, 3) out of bounds for length <union of 30 residue classes (mod 144)> gap> S := Union(S,S^T); < 42 ( 144)java.lang.StringIndexOutOfBoundsException: Index 39 out of bounds for len gap> S := Union(S,S^T); <union of 172 residue classes (mod 432)> gap> S := Union(S,S^T); <union of 676 residue classes (mod 1296)> gap> S := Union(S,S^T); <union of 810 residue classes (mod 1296)> gap> S := Union(S,S^T); <union of 2638 residue classes (mod 3888)> gap> S := Union(S,S^T); <union of 33 residue classes (mod 48)> gap> S := Union(S,S^T); <union of 33 residue classes (mod 48)> gap> Union(S,ResidueClass(Integers,3,0)); # Et voila ... Integers ]]> </Example> Further similar computations are shownjava.lang.StringIndexOutOfBoundsException: Ind Section <Ref Label="sec:CollatzImagesAndPreImages"/>. <P/> Enter <C>AssignGlobals(LoadRCWAExamples().CollatzMapping);</C> norder variablesdefined n section. </Section> <!-- ############################\ <Section Label="sec:MatthewsLeighExamples"> <Heading>Two examples by Matthews and Leigh</Heading> In <Cite Key="MatthewsLeigh87"/>, K. R. Matthews and .nbsp;eighhave shown twotrajectoriesof the following (surjective, but not injective) mappings are acyclic (mod <M>x</M>) and divergent: <Example> <![CDATA[ gap> x := Indeterminate(GF(4),1);; SetName(x,"x"); gap> R := PolynomialRing(GF(2),1); GF(2)[x] gap> ML1 := RcwaMapping(R,x,[[1,0,x],[(x+1)^3,1,x]]*One(R));; gap> ML2 := RcwaMapping(R,x,[[1,0,x],[(x+1)^2,1,x]]*One(R));; gap> Display(ML1); Rcwa mapping of GF(2)[x] with modulus x / | P/x if P in 0(x) P |-> < ((x^3+x^2+x+1)*P + 1)/x if P in 1(x) | \ gap> Display(ML2); Rcwa / | P/x if P in 0(x) P|><(x^+)*P +1/xifP 1x | \ gap> List([ML1,ML2],IsSurjective); [ true, true ] java.lang.StringIndexOutOfBoundsException: Range [11, 3) out of bounds for lengt [ false, false ] gap> traj1 := Trajectory(ML1,One(R),16); [ 1, x^2+x+1, x^4+x^2+x, x^3+x+1, x^5+x^4+x^2, x^4+x^3+x, x^3+x^2+1, x^5+x^2+1, x^7+x^6+x^5+x^3+1, x^9gap List(ycsLength; x^11+x^10+x^8+x^7+x^6+x^5+x^2, x^10+x^9+x^7+x^6+x^5+x^4+x, x^9+x^8+x^6+x^ 22, 21, 22, 22, , ^13x^12+^+x8x^x^+^,x^2+^+x10+^77+^6+x5+^3] gap> traj2 := Trajectory [ x^3+x+1, x^4+x+1, x^5+x^3+gap>u2nu18^462;#This java.lang.StringIndexOutOfBoundsExce x^6+x^4+x^3+x+1, x^7+x^4+x^3+x+1, x^8+x^6+x^5+x^4+x^3+x+1, x^9+x^6+x^3+x+1, x^10+x^8 x^9+x^6+x^] x^11+x^8+x^7+x^5+x^4+x^3+x^2+x+1, x^12+x^10+x^9+x^8+x^7+x^5+1, x^13+x^10+ x^13+x^11+We observethat ourcommutators various java.lang.StringIndexOutOfBoundsExce x^12+x^10+x^9+x^7+x^6+x^4+x^3+x+1 ] ]]> </Example> The pattern which Matthews and Leigh used to show the java.lang.StringIndexOutOfBoundsExce above trajectories can be recognized easily by <java.lang.StringIndexOutOfBoundsExceptio Markov chains with the two states 0 mod <M>x</M> and 1 mod <M>x</M>: <Example> <![CDATA[ ap> := Trajectory(L1One()400x;; gap> traj2modx := Trajectory(ML2,(x^3+x+1)*One(R),600,x);; gap> List(traj1modx{[1..150]},val->Position([Zero(R),One(R)],val)-1); [ 1, 1, 0, 1, 0, 0, 1, 1, 1, 1, 0, 0, 1, 1, 0, 0, 0, 0, 1, 1, 1, 1, 1, 1, 1, 1, 0, 0, 0, 0, 1, 1, 1, 1, 0, 0, 0, 0, 0, 0, 0, 0, 1, 1,GF2)[x] gap : (x^+11,^+1; 1, 1, 1, 1, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 1, gap> c := ClassTransposition(1x,0,^2+x 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 1, gapList(traj2modx{[1..150]},val->Position(Zero(R)O(R),val)1; [ 1, 1, 1, 1, 0, 1, 1, 1, 1, 1, 1, 1, 0, 1, 0, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 0, 1, 0, 1, 0, 1, 0, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, , 0 1 0 1 ,1,1 1 ,1 ,1 , ,1 ,1 ,1, ,, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1 ] ]]> </Example> What is important here are the lengths of the intervals between two changes from one state to the other: <Example> <![CDATA[ gap> ChangePoints := l->Filtered([1..Length(l)-1],pos->l[pos]<>l[pos+1]);; gap> Diffs := l->List([1..Length(l)-1],pos->l[pos+1]-l[pos]);; gap> Diffs(ChangePoints(traj1modx)); # The pattern in the first ... [ 1, 1, 2, 4, 2, 2, 4, 8, 4, 4, 8, 16, 8, 8, 16, 32, 16, 16, 32, 64, 32, true 32, 64 ] gap> Diffs(ChangePoints(traj2modx)); # ... and in the second example. [ 1, 7, 1, 1, 1, 13, 1, 1, 1, 1, 1, 1, 1, 25, 1 1, 1, 1, 1, 1, 1, 49, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, infiniteorder Forexample, <Example> 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, <[ 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 193, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1,] 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1 ] gap> Diffs(ChangePoints(last)); # Make this a bit more obvious. [ 1, 3, 1, 7, 1, 15, 1, 31, 1, 63, 1 ] ]> </Example> This looks clearly acyclic, thus the trajectories diverge. Needless to say however that this computational evidence does not replace gens : GeneratorsOfGroupF); sheds a light on the idea behind it. <P/> Enter <C>AssignGlobals(LoadRCWAExamples().MatthewsLeigh);</C> in order to assign the global variables defined in this section. </Section> <!-- #################################################################### --> <Section Label="sec:OrdersOfCommutators"> <Heading>Orders of commutators</Heading> java.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0 <Example> <![CDATA[ gap> u := RcwaMapping([[3,0,5],[9,1,5],[3,-1,5],[9,-2,5],[9,4,5]]);; gap> IsTame(u);; java.lang.StringIndexOutOfBoundsException: Index 2 out of bounds for length 0 ofZ with java.lang.StringIndexOutOfBoundsException: Index 41 out of bounds for len / | 3n/5 if n in 0(5) | (java.lang.StringIndexOutOfBoundsException: Range [22, 20) out of bounds for le n |-> < (3n-1)/5 if n in 2(5) | (9n-gap =NormedWords(,) [ab*] \ ]]> </Example> We would like to compute the order of <M>[u,n \mapsto n + k]</M> and <M>[u^2,n \mapsto n + k]</M> for different values of <M>k</M>: <Example> <![CDATA[ java.lang.StringIndexOutOfBoundsException: Range [21, 3) out of bounds for lengt gap> l := Filtered([0..100],k->IsTame(Comm(u,nu^k))); [ 0, 2, 3, 5, 6, 9, 10, 12, 13, 15, 17, 18, 20, 21, 24, 25, 27, 28, 30, 32, 33, 35, 36, 39, 40, 42, 43, 45, 47, 48, 50, 51, 54, 55, 57, 58, 60 ,63,65,66 69,70,7273,7577,78, 0,81 ,85,87, ,90,92 , 95, 96, 99, 100 ] java.lang.StringIndexOutOfBoundsException: Range [20, 3) out of bounds for lengt [1, 6, 5, 3, 5, 5, , , 7,infinity,, 7 ,3, java.lang.StringIndexOutOfBoundsException: Range infinity, 3, 5, 7, infinity, 7, infinity, 3, 5, 5, 3, 5, infinity, infinity, infinity, 5, 3, 5, 5, 3, infinity, 7, infinity, 7, 5, 3, ab*c*a*b**a**,ab*a**a*c**,ab***cc*c*b*,a*b*c**cb*c*a*c infinity, infinity, infinity, 5, 3, 5, 5, 3 ] gap> u2 := u^2; <wild rcwa permutation of Z with modulus 25> gap> java.lang.StringIndexOutOfBoundsException: Range [21, 13) out of bounds for l [ 15 ] gap> Order(Comm(u2,nu^15)); infinity gap> u2nu17 := Comm(u2,nu^17); <rcwa permutation of Z with modulus 81> gap> cycs := ShortCycles(u2nu17,[-100..100],100);; gap> List(cycs,Length); [ 72, 73, 72, 72, 72, 73, 72, 72, 73, 72, 72, 73, 72, 72, 73, 72, 72, 73, 72, 72, 73, 72, 72 ] gap> Lcm(last); 5256 gap> u2nu17^5256; IdentityMapping( Integers ) gap> u2nu18 := Comm(u2,nu^18); <rcwa permutation of Z with modulus 81> gap> cycs := ShortCycles(u2nu18,[-100..100],100);; gap> List(cycs,Length); [ 21, 22, 22, 22, 21, 22, 22, 21, 22, 22, 21, 22, 21, 22, 22, 21, 22, 22, 21, 22, 22, 21, 22 ] gap> Lcm(last); 62 x^+x^4x^x+1,x5+^22+(2)*, ^55+x^+(22)2x+(^)2 java.lang.StringIndexOutOfBoundsExcept IdentityMapping( Integers ) gap> List([Comm(u2,nu^20),Comm(u2,nu^25),Comm(u2,nu^30)],Order); [ 29, 9, 15 ] ]]> </Example> We that the prime factors of these orders are not all <Q>very small</Q>. <P/> Enter <C in order to assign the global variables defined in this section. </Section> <!-- #################################################################### --> <Section Label="sec:OddNumberOfGens_FiniteOrder"> <Heading> An infinite subgroup of CT(GF(2)[x]) with many torsion elements </Heading> In this section, we have a look at the following subgroup of CT(GF(2)[x]): <Example> <![CDATA[ gap> x := Indeterminate(GF(2));; SetName(x,"x"); gap> R := PolynomialRing(GF(2),1); GF(2)[x] a: ,,) gap> b := ClassTransposition(0,x^2+1,1,x^2+1);; gap> c := ClassTransposition(1,x,0,x^2+x);; gap> G := Group(a,b,c); <rcwa group over GF(2)[x] with 3 generators> gap> Display(G); Rcwa group over GF(2)[x], generated by [ 2x:P >P +Z()^0 Rcwa permutation of GF(2)[x] with modulus x^2+1, of order 2 / | P + 1 if P in 0(x^2+1) U 1(x^2+1) P |-> < P if P in x(x^2+1) U x+1(x^2+1) | \ Rcwa permutation of GF(2)[x] with modulus x^2+x, of order 2 / | (x+1)*P + x+1 if P in 1(x) P |-> < (P + x+1)/(x+1) if P in 0(x^2+x) | P if P in x(x^2+x) \ ] ]]> We can easily find 2 normal subgroups of <C>G</C>: <Example> <![CDATA[ gap> N1 := Subgroup(G,[a*b,a*c]); <rcwa group over GFjava.lang.StringIndexOutOfBoundsException: Range [19, 20) out of gap> IsNormalfree onthegenerators[ ,b> gap> (,SPermuted(S(,4)OnTuplesF) gap> Index(G,N1); 2 gap GN1 Group([ (1,2), (1,2), (1,2) ]) gap> N2 := Subgroup(G,[a*b*c,a*c]);; gap> IsNormal(G,N2); true gap> IsSubgroup(N1,N2); false ]]> </Example> Products of even numbers of generators of <C>G</C> may have infinite order. For example,We with something onecan tryingto <>transfer/ an <Example> <![CDATA[ gap> Order(a*b); 2 gap> Order(a*c); infinity gap> Order(b*c); infinity ]]> </Example> We would like to have a look at orders of products of odd numbers of generators. In order to restrict our considerations to <Q>essentially different</Q> products (as far as we can easily do this), we use the following auxiliary function: <Listing Type="GAP code"> <![CDATA[ NormedWords := function ( F, lng ) localofnbsp;M&ZZ_(2)}<M> gens := GeneratorsOfGroup(F); tuples := EnumeratorOfTuples([1..3],lng); words := []; for w in tuples do if (w[1] = 1 or not 1 in w) andPositionSublistw[,] =fail and PositionSublist(w,[2,2]) = fail and PositionSublist(w,[3,3]) = fail and PositionSublist(w,[2,1]) = fail and w[1] < w[lng] and w{[1,lng]} <> [1,2] and (w{[1..3]} = [1,2,3] or PositionSublist(w,[1,2,3]) = fail) Add(w) fi od; words := List(words,word->Product(List(word,i->gens[i]))); return words; java.lang.StringIndexOutOfBoundsException: Index 5 out of bounds for length 4 ]]> </Listing> Now let's compute the possible orders of products of 3, 5, 7 or 9 generators: <Log> <![CDATA[ gap gap> phi := EpimorphismByGenerators(F,G); [ a, b, c ] -> [ ClassTransposition(0,x,1,x), ClassTransposition(0,x^2+1,1,x^2+1), ClassTransposition(1,x,0,x^2+x)|(+1/ if in(8) gap> B3 := NormedWords(F,3); [ a*b*c ] gap> B3 := List(B3,g->g^phi); [ <rcwa permutation of GF(2)[x] with modulus x^3+x> ] gap> List(B3,Order); [ 20 ] gap> B5 := NormedWords(F,5); [ a*b*c*a*c, a*b*c*b*c ] gap> B5 := List(B5,g->g^phi); [ <rcwa permutation of GF(2)[x] with modulus x^3+x>, <rcwa permutation of GF(2)[x] with modulus x^4+x^3+x^2+x> ] gap> List(B5,Order); [ 12, 12 ] gap> B7 := NormedWords(F,7); [ a*b*c*a*c*a*c, a*b*c*a*c*b*c, a*b*c*b*c*a*c, a*b*c*b*c*b*c ] gap> B7 := List(B7,g->g^phi); [ <rcwa permutation of GF | 2() <rcwa|n+4 nin1) <rcwa permutationn|- n- java.lang.StringIndexOutOfBoundsException: Range [9, 20) out <rcwa permutation of GF(2)[x] with modulus x^5+x> ] gap> List(B7,Order); [ 12, 12, 12, 30 ] gap> B9 := NormedWords(F,9); [ a*b*c*a*b*c*a*b*c, a*b*c*a*c*a*c*a*c, a*b*c*a*c*a*c*b*c, a*b*c*a*c*b*c*a*c, a*b*c*a*c*b*c*b*c, a*b*c*b*c*a*c*a*c, a*b*c*b*c*a*c*b*c, a*b*c*b*c*b*c*a*c, a*b*c*b*c*b*c*b*c ] gap> B9 := List(B9,g->g^phi);; gap> List(B9,Order); [ 20, 4, 30, 12, 42, 30, 4, 42, 12 ] ]]> </Log> Enter <C>AssignGlobals(LoadRCWAExamples<Chapter Label="Functions"> <H>java.lang.StringIndexOutOfBoundsException: Range [20, 19) out of bounds for len < <!-- #################################################################### --> < Labels:"> <Heading>An abelian rcwa group over a polynomial ring enter a -abelianwildrcwa over ()<M>/>: </> java.lang.StringIndexOutOfBoundsException: Index 37 out of bounds for length 9 ((4,);SetNamex"x"; R :=PolynomialRingGF()1; GF(2^2)[x] gapInclude Label"ycleIndexWreathProductImprimitiveAction" gap> p := x^2 + x + e;; q := x^2 + e;; gap> r := x^2 + x + Z(4);; s := x^2 + x + Z(4)^2;; gap> cg := List( AllResidues(R,x^2), pol -> [ p, p * pol mod q, q ] );; gap> ch := List( AllResidues(R,x^2), pol -> [ r, r * pol mod s, s ] );; gap> g := RcwaMapping( R, q, cg ); <rcwa mapping of GF(2^2)[x] with modulus x^2+1> gap> h := RcwaMapping( R, s, ch ); <rcwa mapping of java.lang.StringIndexOutOfBoundsException: Range [17, 19) out of bo gap> List([g,h],IsTame); [ false, false ] gap> G := Group(g,h); <rcwa group over GF(2^2)[x] with 2 generators> gap> IsAbelian(G); true gap> IsTame(G); false ]]> </Example> It is easy to see|n-31 ifnin () of <C>G</C> are finite. <P/> Now we compute| make some statistics of the orbits of <C>G</C> containing polynomials of degree less than 4: <Example> <![CDATA[ gap> orb := Orbit(G,x^5); [ x^5, x^5+x^4+x^2+1, x^5+x^3+x^2+Z(2^2)*x+Z(2)^0, x^5+x^3, x^5+x^4+x^3+x^2+Z(2^2)^2*x+Z(2^2)^2, x^5+x, x^5+x^4+x^3, x^5+x^2+Z(2^2)^2*x, x^5+x^4+x^2+x, x^5+x^3+x^2+Z(2^2)^2*x+Z(2)^0, x^5+x^4+Z(2^2)*x+Z(2^2), x^5+x^3+x, x^5+x^4+x^3+x^2+Z(2^2)*x+Z(2^2), x^5+x^4+x^3+x+1, x^5+x^2+Z(2^2)*x, x^5+x^4+Z(2^2)^2*x+Z(2^2)^2 ] gap> H := Action(G,orb); Group([ (1,2,4,7,6,9,12,14)(3,5,8,11,10,13,15,16), (90()76(512) (), () ()80),532, gap> IsAbelian(H); # check ... true gap> IsCyclic(Hjava.lang.StringIndexOutOfBoundsException: Index 60 out of bounds f false gap> Exponent(H); 8 gap> Collected(List(ShortOrbits(G,AllResidues(R,x^4),100),Length)); ), 238(512), 254(512), ]]> </Example> Changing the generators a little changes the structure of the group and its action on the underlying ring a lot: <Example> <![CDATA[ gap> cg[1][2] := cg[1][2] + (x^2 + e) * p * q;; gap> ch[7][2] := ch[7][2] + x * r * s;; gap> g := RcwaMapping( R, q, cg );; h := RcwaMapping( R, s, ch );; gap> G := Group(g,h); <rcwa group over GF(2^2)[x] with 2 generators> gap> IsAbelian(G); false gap> Support(G); GF(2^2)[x] \ [ 1, Z(2^2), Z(2^2)^2 ] gap> orb := Orbit(G,Zero(R));; gap> Length(orb); 87 gap> StructureDescription(Action(G,orb)); "A87" gap> Collected(List(orb,DegreeOfLaurentPolynomial)); [ [ -infinity, 1 ], [ 1, 2 ], [ 2, 4 ], [ 3, 16 ], [ 4, 64 ] ] gap> S := AllResidues(R,x^6);; gap> orbs := ShortOrbits(G,S,-1:finite);; gap> List(orbs,Length); [ 87, 1, 1, 1, 2, 2, 2, 2, 2, 4, 4, 4, 20, 4, 12, 4, 20, 4, 4, 12, 8, 8, 48, 48, 16, 8, 8, 56, 8, 88, 8, 8, 8, 400, 16, 48, 16, 16, 16, 80, 16, 16, 16, 96, 32, 192, 32, 16, 16, 416, 16, 48, 16, 16, 880, 16, 16, 16, 16, 16, 16, 16, 16, 16, 848, 16, 16, 32, 16, 16, 16, 16, 16, 16, 16 ] gap> Position(last,880); 55 gap> Set(orbs[55],DegreeOfLaurentPolynomial); # all elm's have same degree [ 5 ] gap> H := Action(G,orbs[55]);; gap> IsPrimitive(H,MovedPoints(H)); false gap> List(Blocks(H,MovedPoints(H)),Length); [ 110, 110, 110, 110, 110, 110, 110, 110 ] ]]> </Example> Enter <C>AssignGlobals(LoadRCWAExamples().AbelianGroupOverPolynomialRing);</C> in order to assign the global variables defined in this section. </Section> <!-- #################################################################### --> <Section Label="sec:CheckingForSolvability"> <Heading>Checking for solvability</Heading> Presently there is no general method available for testing wild rcwa groups for solvability. However, sometimes the question for solvability can be answered anyway. In the example below, the idea is to find a subgroup <A>U</A> which acts on a finite set <A>S</A> of integers, and which induces on <A>S</A> a non-solvable finite permutation group: <Example> <![CDATA[ ]); gap> b := RcwaMapping([[3,0,2],[3,13,4],[3,0,2],[3,-1,4]]);; gap> G := Group(a,b);; gap> ShortOrbits(Group(Comm(a,b)),[-10..10],100); [ [ -10 ], [ -9 ], [ -30, -21, -14, -13, -11, -8 ], [ -7 ], [ -6 ], [ -12, -5, -4, -3, -2, 1 ], [ -1 ], [ 0 ], [ 2 ], [ 3 ], [ 4, 5, 6, 7, 10, 15 ], [ 8 ], [ 9 ] ] gap> S := [ 4, 5, 6, 7, 10, 15 ];; gap> Cycle(Comm(a,b),4); [ 4, 7, 10, 15, 5, 6 ] gap> elm := RepresentativeAction(G,S,Permuted(S,(1,4)),OnTuples); <rcwa permutation of Z with modulus 81> gap> List(S,n->n^elm); [ 7, 5, 6, 4, 10, 15 ] gap> U: Group(Comm(a,),elm) <rcwa group over Z with 2 generators> gap> Action(U,S); Group([ (1,4,5,6,2,3), (1,4) ]) gap> IsNaturalSymmetricGroup(last); true ]]> </Example> Thus the subgroup <A>U</A> induces on <A>S</A> a natural symmetric group of degree 6. Therefore the group <A>G</A> is not solvable. We conclude this example by factoring the group element <A>elm</A> into 482512) 498512,484512, 500512 486) () <Example> <![CDATA[ gap> F := FreeGroup("a","b"); <free group on the generators [ a, b ]> gap> RepresentativeActionPreImage(G,S,Permuted(S,(1,4)),OnTuples,F); a^-2*b^-2*a*b*a^-1*b*a*b^-2*a gap> a^-2*b^-2*a*b*a^-1*b*a*b^-2*a = elm; true ]]> </Example> Enter <C>AssignGlobals(LoadRCWAExamples().CheckingForSolvability);</C> in order to assign the global variables/Section </Section> <!-- #################################################################### -->="sec:o <Section Label="sec:LocalExample"> <Heading>Some examples over (semi)localizations of the integers</Heading> We start with something one can observe when trying to <Q>transfer</Q> an rcwa mapping from the ring of integers to one of its localizations: <Example> <![CDATA[ gap> agap>[.4]i>ai) gap> IsBijective(a); true gap> a2 := LocalizedRcwaMapping(a,2); <gap e2: [1,][,0],1,,][1,,], gap> IsSurjective(a2); # As expected true gap> IsInjective(a2); # Why not?? false gap> 0^a2; 0 gap> (1/3)^a2; # That's the reason! 0 ]]> </Example> The above can also be explained easily by pointing out that the modulus of the inverse of <C>a</C> is 3, and that 3 is a unit of <M>&ZZ;_{(2)}</M>. Moving to <M>&ZZ;_{(2,3)}</M> solves this problem: <Example> <![CDATA[ gap> a23 := SemilocalizedRcwaMapping(a,[2,3]); <rcwa mapping of Z_( 2, 3 ) with modulus 4> gap> IsBijective(a23); true ]]> </Example> We get additional finite cycles, e.g.: <Example> <![CDATA[ gap> List(ShortOrbits(Group(a23),[0..50]/5,50),orb->Cycle(a23,orb[1])); [ [ 0 ], [ 1/5, 2/5, 3/5 ], [ 4/5, 6/5, 9/5, 8/5, 12/5, 18/5, 27/5, 19/5, 13/5, 11/5, 7/5 ], [ 1 ], [ 2, 3 ], [ 14/5, 21/5, 17/5 ], [ 16/5, 24/5, 36/5, 54/5, 81/5, 62/5, 93/5, 71/5, 52/5, 78/5, 117/5, 89/5, 68/5, 102/5, 153/5, 116/5, 174/5, 261/5, 197/5, 149/5, 113/5, 86/5, 129/5, 98/5, 147/5, 109/5, 83/5, 61/5, 47/5, 34/5, 51/5, 37/5, 29/5, 23/5 ], [ 4, 6, 9, 7, 5 ] ] gap> List(last,Length); [ 1, 3, 11, 1, 2, 3, 34, 5 ] gap> List(ShortOrbits(Group(a23),[0..50]/7,50),orb->Cycle(a23,orb[1])); [ [ 0 ], [ -1/7, 1/7 ], [ 2/7, 3/7, 4/7, 6/7, 9/7, 5/7 ], [ 1 ], [ 2, 3 ], [ 4, 6, 9, 7, 5 ] ] gap> List(last,Length); [ 1, 2, 6, 1, 2, 5 ] ]]> </Example> However the structure of a group with prime set <M>\mathbb{P}</M> remains invariant under the <Q>transfer</Q> from &ZZ; to <M>&ZZ;_{(\mathbb{P})}</M>. <P/> <Q>Transferring</Q> a non-invertible rcwa mapping from the ring of integers to some of its (semi)localizations can also turn it into an invertible one: < <![CDATA[ gap> v := RcwaMapping([[6,0,1],[1,-7,2],[6,0,1],[1,-1,1], > (1.20,->(i); gap> Display(v); Rcwa mapping of Z with modulus 8 / | 6n if n in 0(2) | n-1 if n in 3(4) n |-> < (n-7)/2 if n in 1(8) | (n+1)/2 if n in 5(8) | \ gap true gap> IsSurjective(v); false Image()java.lang.StringIndexOutOfBoundsException: Index 14 out of bounds for len Z \ 4(12) U 8(12) gap> Difference(Integers,last); 4(12) U 8(12) gap> v2 := LocalizedRcwaMapping(v,2); < mappingofZ_(2 > gap> IsBijective(v2); true gap> Display(v2^-1); Rcwa permutation of Z_( 2 ) with modulus 4 / | 1/3 n / 2 if n in 0(4) | 2 n + 7 if n in 1(4) n |-> < n + 1 if n in 2(4) | 2 n - 1 if n in 3(4) \ gap> S := ResidueClass(Z_pi(2),2,0);; l := [S];; gap> for i in [1..10] do Add(l,l[Length(l)]^v2); od; gap> l; # Visibly v2 is wild ... [ 0(2), 0(4), 0(8), 0(16), 0(32), 0(64), 0(128), 0(256), 0(512), 0(1024), 0(2048) ] gap> w2 := RcwaMapping(Z_pi(2),[[1,0,2],[2,-1,1],[1,1,1],[2,-1,1]]);; v2w2 =Commv2);v2w2^-;java.lang.StringIndexOutOfBoundsException: Index 36 out of b gap> Display(v2w2); Rcwa permutation of Z_( 2 ) with modulus 8 / | 3 n if n in 2(4) | n + 4 if n in 1(8) n |-> < n - 4 if n in 5(8) | n if n in 0(4) U 3(4) | \ ]]> </Example> Again, viewed as an rcwa mapping of the integers the commutator given at the end of the example would not be surjective. <P/> Enter <C>AssignGlobals(LoadRCWAExamples().Semilocals);</C> in order to assign the global variables defined in this section. </Section> <!-- #################################################################### --> <Section Label="sec:Twisting257CyclesToModulus32"> <Heading> Twisting 257-cycles into an rcwa mapping with modulus 32 </Heading> We define an rcwa mapping <A>x</A> of order 257 with modulus 32. The easiest way to construct such a mapping is to prescribe a transition graph and then to assign suitable affine mappings to its vertices. <java.lang.StringIndexOutOfBoundsException: Index 8 out of bounds for length 0 <![CDATA[ gap> x_257 := RcwaMapping( > [[ 16, 2, 1], [ 16, 18, 1], [ 1, 16, 1], [ 16, 18, 1], > ], 1,16,1,[16,18 ]java.lang.StringIndexOutOfBoundsException: Index 67 out of bound > 1 16 1,[16 ,] [ 1 1,[16,, 1, > [ 1, 16, 1], [ 16, 18, 1], [ 1, 16, 1], [ 16, 18, 1], > [ 1, 0, 16], [ 16, 18, 1], [ 1,-14, 1], [ 16, 18, 1], > [ 1,-14, 1], [ 16, 18, 1], [ 1,-14, 1], [ 16, 18, 1], > [ 1,-14, 1], [ 16, 18, 1], [ 1,-14, 1], [ 16, 18, 1], > [ 1,-14, 1], [ 16, 18, 1], [ 1,-14, 1], [ 1,-31, 1]]);; gap> Order(x_257);; Display(x_257:CycleNotation:=false); Rcwa permutation of Z with modulus 32, of order 257 / | 16n+18 if n in 1(2) \ 31(32) | n+161() 89) | 12(32) U 14(32) | n-14 if n in 18(32) U 20(32) U 22(32) U 24(32) U 26(32) U n |-> < 28(32) U 30(32) | 16n+2 if n in 0(32) | n/16 if n in 16(32) | n-31 if n in 31(32) | \ gap> Display(x_257); Rcwa permutation of Z with modulus 32, of order 257 (0(), (512,18() 4512, 20() 6512,22(12, 8(512), 24(512), 10(512), 26(512), 12(512), 28(512th <>T6/ residueclass(2) 30512,16() 132,34(), 50(12, 36512,512, 38(512), 54(512), 40(512), 56(512), 42(512), 58(512), 44(512), , ,82512) 68(512), 84(512), 70(512), 86(512), 72(512), 88(512), 74(512), 90(512), 76(512), 92(512), 78(512), 94(512), 80(512), 5(32), 98(512), 114(512), 100(512), 116(512), 102(512), 118(512), 104(512), 120(512), 106(512), 122(512), 108(512), 124(512), ) 126512, 112) 7(),130512),146() 132(512), 148(512), 134(512), 150(512Integers,,)S)java.lang.StringIndexOutOfBoundsE 138(512), 154(512), 140(512), 156(512), 142(512), 158(512), 144(512), 9(32), 162(512), 178(512), 164(512), 180(512), 166(512), 182(512), 168(512), 184(512), 170(512), 186(512), 172(512), 188(512), 174(512), 190(512), 176(512), 11(32), 194(512), 210(512), 196(512), 212(512), 198(512), 214(512), 200(512), 216(512), 202(512), 218(512), 204(512), 220(512), 206(512), 222(512), 208(512), 13(32), 226(512), 242(512), 228(512), 244(512), 230(512), 246(512), 232(512), 248(512), 512,236(12,252(),(512, 254512),java.lang.StringIndexOutOfBoundsException: Index 6 240(512), 15(32), 258(512), 274(512), 260(512), 276(512), 262(512), 278(512), 264(512), 280(512), 266(512), 282(512), 268(512), 284(512), 270(512), 286(512), 272(512), 17(32), 290(512), 306(512), 292(512), 308(512), 294(512), 310(512), 296(512), 312(512), 298(512), 314(512), 300(512), 316(512), 302(512), 318(512), 304(512), 19(32), 322(512), 338(512), 324(512), 340(512), 326(512), 342(512), 328(512), 344(512), 330(512), 346(512), 332(512), 348(512), 334(512), 350(512), 336(512), 21(32), 354(512), 370(512), 356(512), 372(512), 358(512), 374(512), 360(512), 376(512), 362(512), 378(512), 364(512), 380(512), 366(512), 382(512), 368(512), 23(32), 386(512), 402(512), 388(512), 404(512), 390(512), 406(512), 392(512), 408(512), 394(512), 410(512), 396(512), 412(512), 398(512), 414(512), 400(512)0() 420(512), 436(512), 422(512), 438(512), 424(512), 440(512), 426(> : ^T 432(512), 27(32), 450(512), 466(512),03) 5(9 79)U() 454(512), 470(512), 456(512), 472(512), 458(512), 474(512), 460(512), 476(512), 462(512), 478(512), 464(512), 29(32), 482(512), 498(512), 484(512), 500(512), 486(512), 502(512), 488(512), 504(512), 490(512), 506(512), 492(512), 508(512), 494(512), 510(512), 496(512), 31(32) ) gap> Length(Cycle(x_257,0)); 257 ]]> </Example> Enter <C>AssignGlobals(LoadRCWAExamples().LongCyclesOfPrimeLength);</C> in java.lang.StringIndexOutOfBoundsException: Index 8 out of bounds for length 8 </Section> <!-- #################################################################### -->####### <Section Label="sec:ModuliOfPowers"> <Heading> The behaviour of the moduli of powers </Heading> We give some examples of how the series of the moduli of powers of a given rcwa mapping of the integers can look like. <Example> <![CDATA[ a : RcwaMapping(30,02][,1,]); gap> List([0..4],i->Modulus(a^i)); [ 1, 4, 16, 64, 256 ] gap> e1 := RcwaMapping([[1,4,1],[2,0,1],[1,0,2],[2,0,1]]);; gap> e2() U4)U () > [1,4,1],[2,0,1],[1,0,1],[1,0,1]]);; gap> List([e1,e2],Order); [infinity gap> List([1..20],i->Modulus(e1^i)); [ 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4,<nion of784 classesmod 1215> gap> List([1..20],i->Modulus(e2^i)); [ 8, 4, 8, 4, 8, 4, 8, 4, 8, 4, 8, 4, 8, 4, 8, 4, 8, 4, 8, 4 ] gap> Display(e2); Rcwa permutation of Z with modulus 8, of order infinity / | n+4 if n in 0(4) | 2n if n in 1(4) n |-> < n/2 if n in 2(8) | n if n in 3(4) U 6(8) | \ gap> e2^2 = Restriction(RcwaMapping([[1,2,1]]),RcwaMapping([[4,0,1]])); true gap> g:=RcwaMapping([[2,2,1],[1, 4,1],[1,0,2],[2,2,1],[1,-4,1],[1,-2,1]]);; gap> h:=RcwaMapping([[2,2,1],[1,-2,1],[1,0,2],[2,2,1],[1,-1,1],[1, 1,1]]);; gap> List([0..7],i->Modulus(g^i)); [ 1, 6, 12, 12, 12, 12, 6, 1 ] gap> List([1..18],i->Modulus((g^3*h)^i)); [ 12, 6, 12, 12, 12, 6, 12, 6, 12, 12, 12, 6, 12, 6, 12, 12, 12, 6 ] gap> u := RcwaMapping([[3,0,5],[9,1,5],[3,-1,5],[9,-2,5],[9,4,5]]);; gap> List([0..3],i->Modulus(u^i)); [ 1, 5, 25, 125 ] gap> v6 := RcwaMapping([[-1,2,1],[1,-1,1],[1,-1,1]]);; gap> List([0..6],i->Modulus(v6^i)); [ 1, 3, 3, 3, 3, 3, 1 ] gap> w8 := RcwaMapping([[-1,3,1],[1,-1,1],[1,-1,1],[1,-1,1]]);; gap> List([0..8],i->Modulus(w8^i)); [ 1, 4, 4, 4, 4, 4, 4, 4, 1 ] gap> z := RcwaMapping([[2,1,1],[1, 1,1],[2,-1,1],[2, -2,1], > [1,6,2],[1, 1,1],[1,-6,2],[2, 5,1], > [1,6,2],[1, 1,1],[1, 1,1],[2, -5,1], > [1,0,1],[1,-4,1],[1, 0,1],[2,-10,1]]);; gap> IsBijective(z); true gap> List([0..25],i->Modulus(z^i)); [ 1, 16, 32, 64, 64, 128, 128, 128, 128, 128, 128, 256, 256, 256, 256, 256, 256, 512, 512, 512, 512, 512, 512, 1024, 1024, 1024 ] ]]> </Example> Enter <C>AssignGlobals(LoadRCWAExamples().ModuliOfPowers);</C> in order to assign the global variables defined in this section. </Section> <!-- #################################################################### --> <Section Label="sec:CollatzImagesAndPreImages"> <Heading> Images and preimages under the Collatz mapping </Heading> We have a look at the images of the residue class 1(2) under powers of the Collatz mapping. <Example> <![CDATA[ gap> T := RcwaMapping([[1,0,2],[3,1,2]]);; gap> S0 := ResidueClass(Integers,2,1);; S1 : S0^java.lang.StringIndexOutOfBoundsException: Range [16, 17) out of bounds fo 2(3) gap> S2 := S1^T; 1(3) U 8(9) gap [268434943 1619 ],[ 536869887, 2429], [ 1073739775,3644], 2(3) U 4(9) gap> S4 := S3^T; Z \ 0(3) U 5(9) gap> S5 := S4^T; Z \ 0(3) U 7(9) gap> S6 := S5^T; Z \ 70368609926062 20 , ,[140737219852124 5], [ 281474439704249 ,[281474439704249, 4],[562948879408498 , Z \ 0(3) ]]> </Example> Thus the image gets stable after applying the mapping <M>T</M> gap> Trajectory(igma_T^1[,]20; he6th time.Hence M>6/>maps residueclass()surjectively onto the union of the residue classes 1(3) and 2(3), which <M>T</M> stabilizes setwise. Now we would like to determine thepreimagesof 1(3) and 2(3) in 1(2) under <M>T^6</M>. The residue class 1(2) has to be the disjoint union of these sets. <Example> <![CDATA[ gap> U := Intersection(PreImage(T^6,ResidueClass(Integers,3,1)),S0); <unionWhile itseemseasy to make of gapof C>&uscoreT/C>obtaining on < 21 (64) gapAsUnionOfFewClassesU [ 1(64), 5(64), 7(64), 9(64), 21(64), 23(64), 29(64), 31(64), 49(64), 51(64), 59(64) ] gap> AsUnionOfFewClasses(V); [ 3(32), 11(32), 13(32), 15(32), 25(32), 17(64), Rpermutation Z^2 with modulus(1,)(,)java. 37(64), 39(64), 41(64), 53(64), 55(64), 61(64), 63(64) ] gap> Union(U,V) = S0 and Intersection(U,V) = []; # consistency check true ]]> </Example> The images of the residue class 0(3) under powers of <M>T</M> look as follows: <Example> <![CDATA[ gap> / 0(3) gap> S1 0()U(9) gap> S2 := S1^T; 0(3) U 5(9) U 7(9) U 8(27) gap> S3 := S2^T; <union of 20 residue classes (mod 27) (6 classes)> gap> S4 := S3^T; <union of 73 residue classes (mod 81)> gap> S5 := S4^T; Z \ 10(81) U 37(81) gap> S6 := S5^T; Integers gap> S7 := S6^T; Integers ]]> </Example> Thus every integer is the image of a multiple of 3 nder&;M means it wouldbesufficient toprovethe <M>3n+1</M> conjecture for multiples of 3. We can obtain the corresponding result for multiples of 5 as follows: <Example> <[62,16001] 31 ],[15 4005 , 7 128011 ], gap> S 3 256023 ], 1,512047] 0 ], [0 ] [ 0(5) ] gap> for i in [1..12] do Add(S,S[i]^T); od; gap> for s in java.lang.StringIndexOutOfBoundsException: Range [0, 15) out of bounds 0(5) 0(5) U 8(15) 0(5) U 4(15) U 8(15) 0(5) U 2(15) U 4(15) U 8(15) U 29(45) <union of 73 residue classes (mod 135)> <union of 244 residue classes (mod 405)> <union of 784 residue classes (mod 1215)> <union of 824 residue classes (mod 1215)> <union of 2593 residue classes (mod 3645)> <union of 2647 residue classes (mod 3645)> <union of 2665 residue classes (mod 3645)> <union of 2671 residue classes (mod 3645)> 1(3) U 2(3) U 0(15) gap> Union(S[13],ResidueClass(Integers,3,0)); Integers gap> List(S,Si->Float(Density(Si))); [ 0.2, 0.266667, 0.333333, 0.422222, 0.540741, 0.602469, 0.645267, 0.678189, 0.711385, 0.7262, 0.731139, 0.732785, 0.733333 ] ]]> </Example> Enter <C>AssignGlobals(LoadRCWAExamples().CollatzMapping);</C> in order to assign the global variables defined in this section. </Section> <!-- #################################################################### --> <Section Label="sec:Sigma_T"> <Heading> An extension of the Collatz mapping T to a permutation of <M>&ZZ;^2</M> </Heading> The Collatz mapping <M>T</M> is java.lang.StringIndexOutOfBoundsException: Range [0, 47) <Example> <![CDATA[ gap> T := RcwaMapping([[1,0,2],[3,1,2]]);; gap> Display(T); Rcwa mapping of Z with modulus 2Listing> / | n/2 if n in 0(2) n |-> < (3n+1)/2 if n in 1(2) | \ gap> IsInjective(T); IsSurjective(T); false true gap> PreImages(T,2); [ 1, 4 ] ]]> </Example> Often, dealing with rcwa permutations is easier. Indeed the Collatz mapping&Listing> ways to permutations of <M>![CDATA[ permutation acts a=leveljava.lang.StringIndexOutOfBoundsException: Range [45, 44) Example <![CDATA[ gap> Sigma_T := RcwaMapping( Integers^2, [[1,0],[0,6]], > [[[[2,0],[/> > [[[4,0],[0,3]],[2,1],2], > [[[2,0],[0,1]],[0,0],2], > [[[4,0],[0,3]],[2,1],2], > [[[4,0],[0,1]],[0,0],2], > [[[4,0Example <rcwa mapping of Z^2 with modulus (1,0)Z+(0,6)Z> gap> IsBijective(Sigma_T); true0() () 4,3()) gap> Display(Sigma_T); Rcwa permutation of Z^2 with modulus (1,0)Z+(0,6)>iflevel=1 ;else / | (2m+1,(3n+1)/2) if (m,n) in (0,1)+(1,0)Z+(0,2)Z | (m,n/2) if (m,n) in (0,0)+(1,0)Z+(0,6)Z U (m,n) |-> < (0,2)+(1,0)Z+(0,6)Z | (2m,n/2) if (m,n) in (0,4)+(1,0)Z+(0,6)Z | \ gap> Display(Sigma_T^-1); Rcwa permutation of Z^2 with modulus (2,0)Z+(0,3)Z / | (m,2n) if (m,n) in (0,0)+(1,0)Z+(0,3)Z U | (0,1)+(1,0)Z+(0,3)Z (m,n) |-> < (m/2,2n) if (m,n) in (0,2)+(2,0)Z+(0,3)Z | ((m-1)/2,(2n-1)/3) if (<groupoverZ with 2 > \ ]]> </Example> ow <>3+<M conjectureis equivalentto assertionthat the line <M>n=4</M> is a set (15,9,13)261014(,,12, ], <C>Sigma&uscore;T</C> on the half plane <M>n > 0</M>. <P/> gap H,Size); <Example> <![CDATA[ gap>Sigma_T,[,,75; [ [ 0, 27 ], [ 1, 41 ], [ 3, 62 ], [ 3, 31 ], [ 7, 47 ], [ 15, gap>List(H,; [ 31, 107 ], [ 63, 161 ], [ 127, 242 ], [ 127, 121 ], [ 255, 182 ], [ 255, 91 ], [ 511, 137 ], [ 1023, 206 ], [ 1023, 103 ], [ 2047, 155 ], [ 4095, 233 ], [ 8191, 350 ], [ 8191, 175 ], [ 16383,263 ,[32767,395 , ,593] 131071 ] [ 131071, 445 ], [ 262143, 668 ], [ 262143, 334 ], [ 524286,in theglobalvariablesdefinedinthis [ 1048573, 251 ], [ 2097147, 377 ], [ 4194295, 566 ], [ 4194295, 283 ], [8388591, , [ 33554367, 479 ], [ 67108735, 719 ], [ 134217471, 1079 ], [ 268434943, 1619 ], [ 536869887, 2429 ], [ 1073739775, 3644 ], [ 1073739775, 1822 ], [ 2147479550, 911 ], [ 4294959101, 1367 ], [ 8589918203, 2051 ], [ 17179836407, 3077 ], [ 34359672815, 4616 ], [ 34359672815, 2308 ], [ 68719345630, 1154 ], [ 68719345630, 577 ], [ 137438691261, 866 ], [ 137438691261, 433 ], [ 274877382523, 650 ], [ 274877382523, 325 ], [ 549754765047java.lang.StringIndexOutOfBoundsException: Index [ 1099509530094, 122 integer the<><M>&;.Incontrast, [ 2199019060189, 46 ], [ it seems likely that mosttrajectories of twomappings [ 17592152481515, 53 ], [ 35184304963031, 80 ], [ 35184304963031, 40 ], [ 70368609926062, 20 ], [ 70368609926062, 10 ], [ 140737219852124, 5 ], [281474439704249, ,[281474439704249,4] [562948879408498,2] [ 562948879408498, 1 ], [ 1125897758816997, 2 ], [ 1125897758816997, 1 ], [ 2251795517633995, 2 ], [ 2251795517633995, 1 ] ] gap> Trajectory(Sigma_T^-1,[0,27],20); [ [ 0, 27 ], [ 0, 54 ], [ 0, 108 ], [ 0, 216 ], [ 0, 432 ], [ 0, 864 ], [ 0, 1728 ], [ 0, 3456 ], [ 0, 6912 ], [ 0, 13824 ], [ 0, 27648 ], [ 0, 55296 ], [ 0, 110592 ], [ 0, 221184 ], [ 0, 442368 ], [ 0, 884736 ], [ png"width 372" height = "" [ 0, 14155776 ] ] ]]> /java.lang.StringIndexOutOfBoundsException: Range [10, 9) out of bounds for leng While it seems easy to make conjectures regarding the behaviour of cycles of <C>Sigma&uscore;T</C>, obtaining results on it is apparently hard. We observe however that <C>Sigma&uscore;T</C> can be written as a product of two permutations of <M>&ZZ;^2</M> whose cycles can be described easily: <Example> <![CDATA[ gIntegers^2[10,02],[[40,01],0 0,]java.lang.StringIndexOutOfBoundsException: Index > [[[4,0],[0,1]],[2,-1],2]]); <rcwa mapping of Z^2 with modulus (1,0)Z+(0,2)Z> gap> b := a^-1*Sigma_T; <rcwa permutation of Z^2 with modulus (2,0)Z+(0,3)Z> gap> Display(a); Rcwa permutation of Z^2 with modulus (1,0)Z+(0,2)Z / | (2m,n/2) if (m,n) in (0,0)+(1,0)Z+(0,2)Z (m,n) |-> < (2m+1,(n-1)/2) if (mjava.lang.StringIndexOutOfBoundsException: Range [8, 7) | \ gap> Display(b); Rcwa permutation of Z^2 with modulus (2,0)Z+(0,3)Z / | (m,3n+2) if 5:Z \ 7(2 2532) m/,) (m, in(,)(,)+03Z (m,n) |-> < (0,1)+(2,0)Z+(0,3)Z | (m,n) if (m,n)> | \ ]]> </Example> It is easy to see that both <C>a</C> and <C>b</C> have infinite order. The <small/Q <><M.Sincei Mca/>wehave speak, from <M>(0<>anb/ geq </ Mn\ /ca)/M, we java.lang.StringIndexOutOfBoundsException: Rang A given cycle contains only finitely many points both of whose coordinates are (0)and (1,1) We have a look at an example of a cycle of <Example> <![CDATA[ gap> Trajectory(a,[1000,1000],15); [ [ 1000, 1000 ], [ 2000, 500 ], [ 4000, 250 ], [ 8000, 125 ], [ 16001, 62 ], [ 32002, 31 ], [ 64005, 15 ], [ 128011, 7 ], [ 256023, 3 ], [ 512047, 1 ], [ 1024095, 0 ], [ 2048190, 0 ] [ 4096380, / gap> Trajectory(a^-1,[1000,1000],15); [ [ 1000, 1000 ], [ 500, 2000 ], [ 250, 4000 ], [ 125, 8000 ], [ 62, 16001 ], [ 31, 32002 ], [ 15, 64005 ], [ 7containnbsp; by computing preimages ofnbsp1 [ 3, 256023 ], [ 1, 512047 ], [ 0, 1024095 ], [ 0, 2048190 ], [ 0, 4096380 ], [ 0, 8192760 ], [ 0, 16385520 ] ] ]]> </Example> tisleft aneasyexercise theto the cycles of <C>b</C> look like k :k+; Enter <C>AssignGlobals(LoadRCWAExamples()> until (small; in order to assign the global variables defined in this section. </Section]> <!-- #################################################################### --> <Section Label="sec: <Heading> Finite quotients of Grigorchuk groups </Heading> In this section, we show how to construct finite quotients of the two infinite periodic groups introduced by Rostislav Grigorchuk in <Cite Key="Grigorchuk80"/> with the help of & The free group of rank 2 and the modular group The first of these, nowadays known as <Q>Grigorchuk group</Q>, is in anexample onthe &AP website- <URL>https:even in subgroupwhich s by classtranspositions The &RCWA; package permits a simpler and more elegant construction of the finite quotients of this group: The function <C>TopElement</C> given on the mentioned webpage gets unnecessary, and the function <C>SequenceElement</C> can be simplified as follows: <Listing> <![CDATA[ SequenceElement := function ( r, level ) return Permutation(Product(Filtered([1..level-1],kgap>gamma1 =RepresentativeActionR k->ClassTransposition( 2^(k-1)-1,2^(k+1), 2^k+2^(k-1)-1,2^(k+1))), [0..2^level-1]); ]]> </Listing> The actual constructors for the generators are modified as follows: <Listing> <![CDATA[ a := level -> Permutation(ClassTransposition(0,2,1,2),[0..2^level-1]); b := level -> SequenceElement(0,level); c := level -> SequenceElement(2,level); d := level -> SequenceElement(1,level); ]]> </Listing> All computations given on the webpage can now be done just as with the <Q>original</Q> construction of the quotients of the Grigorchuk group. In the sequel, we construct finite quotients of the second group introduced in <Cite Key="Grigorchuk80"/>: <Example> <![CDATA[ gap> FourCycle := RcwaMapping((4,5,6,7),[4..7]); ( 0(4), 1(4), 2(4), 3(4) ) gap> GrigorchukGroup2Generator := function ( level ) > if level = 1 then return FourCycle; else > return Restriction(FourCycle, RcwaMapping([[4,1,1]])) > * Restriction(FourCycle, RcwaMapping([[4,3,1]])) > * Restriction(GrigorchukGroup2Generator(level-1), > RcwaMapping([[4,0,1]])); > fi; > end;; gap> GrigorchukGroup2 := level -> Group(FourCycle, > GrigorchukGroup2Generator(level));; ]]> </Example> We can do similar things as shown in the example on the &GAP; webpage for the <Q>first</Q> Grigorchuk group: E <![CDATA[ gap> G := List([1..4],lev->GrigorchukGroup2(lev<Example> [ <rcwa group over Z with 2 generators>, <rcwa group over Z with 2 generators>Attr=IsomorphismRcwaGroup" <rcwa group over Z with 2 generators>, <rcwa group over Z with 2 generators> ] gap> H := List([1.</> [ Group([ (1,2,3,4), (1,2,3,4) ]), Group([ (1,2,3,4)(5,6,7,8)(9,10,11,12)(13,14,15,16), (1,5,9,13)(2,6,10,14)(4,8,12,16) ]), <permutation group with 2 generators>, <permutation group with 2 generators> ] gap> List(H,Size); [ 4, 1024, 4294967296, 1329227995784915872903807060280344576 ] gap> List(last,n->Collected(Factors(n))); [ [ [ 2, 2 ] ], [ [ 2, 10 ] ], [ [ 2, 32 ] ], [ [ 2, 120 ] ] ] gap> List(H,NilpotencyClassOfGroup); [ 1, 6, 14, 40 ] ]] </Example> Enter <C>AssignGlobals(LoadRCWAExamplesgap> IsSubset(X1,X2^.) andIsSubset(1X^(F21^-1)) in order> andIsSubset(2X1^2 IsSubsetX2X1^F22-1); </Section> <!-- #################################################################### --> <"> <Heading> Forward orbits of a monoid with 2 generators </Heading> The <M>3n+1</M> > IsSubset(Union(Sinks^1)X2^F2^-1)java.lang.StringIndexOutOfBoundsExcept integer under the Collatz mapping <M>T</M> contains 1. In contrast, it seems likely that <Q>most</Q> trajectories of the two mappings <Alt Only="LaTeX"> <Display> <![CDATA[T_5^\pm: \ \mathbb{Z} \longrightarrow \mathbb{Z}, \ \ \ \ n \ \longmapsto \ \begin{cases} \frac{n}{2} & \may help to understand what is actually tounderstandwhat inthiscalculat f5n \ 1{2}&\text{f \ \ \end{cases}]]> </Display> </Alt> <Alt Only="HTML"><![CDATA[<center> <img src = "t5pm.png" width = "372" height = "61" alt = "T5+/-: Z -> java.lang.StringIndexOutOfBoundsException: Index 1 out of bounds fo </center>]]></Alt> <Alt Only="Text"><Verb><![CDATA[ / | n/2 if n even, T_5+/-: Z -> Z, n |-> < | (5n +/- 1)/2 if n odd \ ]]></Verb></Alt> diverge. However we can show by means of computation that the forward orbit of any positive integer under the action of the monoid generated by the two mappings <M>T_5^-</M> and <M>T_5^+</M> indeed java.lang.StringIndexOutOfBoundsException: Inde First of all, we enter the generators: <Example> <![CDATA[ gap> T5m := RcwaMapping([[1,0,2],[5,-1,2]]);; gap> T5p := RcwaMapping([[1,0,2],[5, 1,2]]);; ]]> </Example> a Mk<M Mr2^)<M there is a product <M>f> CyclicGroup2))java.lang.StringIndexOutOfBoundsException: Index whose restriction to <M>r(2^k)</M> is given by <M>n \mapsto (an+b)/c</M> where <M>c>a</M>: <Example> <![CDATA[ gap> k := 1;; gap> repeat > maps := List(Tuples([T5m,T5p],k),Product); > decr := List(maps,DecreasingOn); > decreasable := Union > Print(k,": ")< Only=HTML"&<Alt> > k := k + 1; > until 1: 0(2) 2: 0(4) 3: Z \ 1(8) U 7(8) 4: 0(4) U 3(16) U 6(16) U 10(16) U 13(16) 5: Z \ 7(32) U 25(32) 6: <union of 48 residue classes (mod 64)> 7: Integers ]]> </Example> Thus <M>k=7</M> serves our purposes. To be sure that for any positive integer <M>n</M> our monoid contains a mapping <M>f</M> such that <M>n^f<n</M>, we still need to check this condition for <Q>small</Q> <M>n</M>. Since in case <M>c>a</M> we have <M>(an+b)/c \geq n</M> if only if <M>n \leq b/(c-a)</M>, we only need to check those <M>n</M> which are not larger than the largest coefficient <M>b_{r(m)}</M> occurring in any of the products under consideration: <Example> <![CDATA[ gap> maxb := Maximum(List(maps,f->Maximum(List(Coefficients(f),t->t[2])))); 25999 gap> small := Filtered([1..maxb],n->ForAll(maps,f->n^f>=n)); [ 1, 7, 9, 11 ] ]]> </Example> This means that except of 1, only for <M>n \in \{{7,9,11\}}</M> there is no product of 7 mappings <M>T_5^\pm</M> which maps <M>n</M> to a smaller integer. We check that also the forward orbits of these three integers contain 1 by successively computing preimages of 1: <Example> <![CDATA[ gap> S := [1];; k := 0;; gap> repeat > S := Union(S,PreImage(T5m,S),PreImage(T5p,S)); > k := k+1; > until IsSubset(S,small); gap> k; 17 ]]> </Example> CAssignGlobals().);<C inorder to theglobal variables efinedinthis. </Section> !-################################## -> <Section Label="sec:F2andPSL2Z"> <Headingjava.lang.StringIndexOutOfBoundsException: Index 9 out of bounds for leng /> free of rank 2embeds RCWA&; - infactit be constructedby transferringthe theso- <QSchottky <Q>(f. <Cite Key=LaHarpe00/, page 27) from PSL(2, (we use the notation from the cited book): < !CDATA gap D: AllResidueClassesModulo(); 0(4), 1(4),2(4), 3() ] gap gamma1 =RepresentativeAction(()java.lang.StringIndexOutOfBoundsException: Inde >Difference(D[],D2); gap> gamma2 := java.lang.StringIndexOutOfBoundsException: Index 28 out of bounds for >Difference(Integers,D3)D]); gap> F2 := Group"NCurses.BrowseGeneric<code></span></h5> <rcwa group over Z with 2 generators> ]]> </Example> We can do some checks: <Example> <![CDATA[ gap> X1 := Union(D{[1,2]});; X2 := Union(D{[3,4]});; gap> IsSubset(X1,X2^gamma1) and IsSubset(X1,X2^(gamma1^-1)) > and IsSubset(X2,X1^gamma2) and IsSubset(X2,X1^(gamma2^-1)); true ]]> </Example> The generators are products of 3 class transpositions, each: <Example> <![CDATA[ gap> Factorization(gamma1); [ ( 0(2), 1(2) ), ( 3(4), 5(8) ), ( 0(2), 1(8) ) ] gap> Factorization(gamma2); [ ( 0(), ()) () 78 ,( 02, 38) ] ] </Example> constructionused by<ef " =for group">to embed free groups of any rank <M>\geq 2</M>. </ We give another only slightly different representation of the free group of rank 2. We verify that it really is one by java.lang.StringIndexOutOfBoundsException: In - ETable-Lemma</E>(ee g Cite =LaHarpe00"> Section II.B.) to the infinite cyclic groups generated by the two generatorsandtothe two sets <>X1</C and<><C as above: < <![CDATA[ gap> r1 := ClassTransposition(0,2,1,2)*ClassTransposition(0,2,1,4);; gap> r2 := ClassTransposition(0,2,1,2)*ClassTransposition(0,2,3,4);; gap> F2 := Group(r1^2,r2^2);; gap> List(GeneratorsOfGroup(F2),IsTame); [ false, false ] gap> IsSubset(X1,X2^F2.1) and IsSubset(X1,X2^(F2.1^-1)) > and IsSubset(X2,X1^F2.2) and IsSubset(X2,X1^(F2.2^-1)); true gap> [Sources(r1),Sinks(r1),Loops(r1)]; # compare with X1 [ [ 0(4) ], [ 1(4) ], [ 0(4), 1(4) ] ] gap> [Sources(r2),Sinks(r2),Loops(r2)]; # compare with X2 [ [ 2(4) ], [ 3(4) ], [ 2(4), 3(4) ] ] gap> IsSubset(X1,Union(Sinks(r1))) and IsSubset(X1,Union(Sinks(r1^-1))) > and IsSubset(X2,Union(Sinks(r2))) and IsSubset(X2,Union(Sinks(r2^-1))); true gap> IsSubset(Union(Sinks(r1)),X2^F2.1) and > IsSubset(Union(Sinks(r1^-1)),X2^(F2.1^-1)); true gap> IsSubset(Union(Sinks(r2)),X1^F2.2) and > IsSubset(Union(Sinks(r2^-1)),X1^(F2.2^-1)); true ]]> </Example> Drawing the transition graphs of <C>r1</C> and <C>r2</C> for modulus 4 may help to understand what is actually done in this calculation. It is easy to see that the group generated by <C>r1</C> and <C>r2</C> is <E>not</E> free: <Example> <![CDATA[ gap> Order(r1/r2); 3 ]]> </Example> The modular group PSL(2,&ZZ;) embeds into CT(&ZZ;) as well. We give an embedding, and check that it really is one by applying the Table Tennis Lemma as above: <Example> <![CDATA[ gap> PSL2Z := > Group(ClassTransposition(0,3,1,3) * ClassTransposition(0,3,2,3), > ClassTransposition(1,3,0,6) * ClassTransposition(2,3,3,6));; gap> List(GeneratorsOfGroup(PSL2Z),Order); [ 3, 2 ] gap> X1 := Difference(Integers,ResidueClass(0,3)); Z \ 0(3) gap> X2 := ResidueClass(0,3); 0(3) gap> IsSubset(X1,X2^PSL2Z.1) and IsSubset(X1,X2^(PSL2Z.1^2)); true gap> IsSubset(X2,X1^PSL2Z.2); true ]]> </Example> A slightly different representation of PSL(2,&ZZ;) can be obtained by using &RCWA;'s general method for <C>IsomorphismRcwaGroup</C> for free products of finite groups: <Example> <![CDATA[ gap> G := Image(IsomorphismRcwaGroup(FreeProduct(CyclicGroup(3), > CyclicGroup(2)))); <wild rcwa group over Z with 2 generators> gap> List(GeneratorsOfGroup(G),Factorization); [ [ ( 0(4), 2(4) ), ( 1(2), 0(4) ) ], [ ( 0(2), 1(2) ) ] ] ]]> </Example> Enter <C>AssignGlobals(LoadRCWAExamples().F2_PSL2Z);</C> in order to assign the global variables defined in this section. <Alt Only="HTML"> </Alt> </Section> <!-- #################################################################### --> </Chapter> <!-- #################################################################### -->
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