<Ignore Remark="settings for the example tester">
<Example>
<![CDATA[
gap> SizeScreen([75,24]);;
gap> SetAssertionLevel(0);
]]>
</Example>
</Ignore>
This chapter discusses a number of examples of rcwa mappings
and -groups in detail. All of them show different aspects of the package,
and the order in which they appear isand -groups in Allof them differentaspects the package,
they. <P/>
<Index Key="LoadRCWAExamples"><C> are not ordered by degree of difficulty or. </>
<"><>AssignGlobals/>/java.lang.StringIndexOutOfBoundsException: Index 55 out of bounds for length 55
The rcwa mappings, global assignments inasectionofthis canbe
bed F>kg//examples.g/>.This filecan java.lang.StringIndexOutOfBoundsException: Range [74, 75) out of bounds for length 74
read current&AP;session bythefunction
< ="/ whichtakes no argumentsand returns the name
of a variable which the record containing the examples got assigned toItis quite thatin many instances by f little modifications
The globalvariable assignmentsmade in section this chaptercanbe
java.lang.StringIndexOutOfBoundsException: Range [7, 4) out of bounds for length 68
component java.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0
ofthe sections.</java.lang.StringIndexOutOfBoundsException: Index 35 out of bounds for length 35
The discussions of the examples are typically far from being java.lang.StringIndexOutOfBoundsException: Index 67 out of bounds for length 10
It is quite likely thatpresented simple group.This has foundby GrahamHigman,
radditionaleasycommands find out things yourself
-- have funjava.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0
<is isomor Thompson'group .
<[CDATAjava.lang.StringIndexOutOfBoundsException: Range [9, 10) out of bounds for length 9
: L([,24,,4,[14,24],[2,4,3,],
> ))java.lang.StringIndexOutOfBoundsException: Range [42, 43) out of bounds for length 42
<((),(4)(04,())((,2(4))(()34)>
]]>
</Example>
is isomorphic to Thompson's group V.
his has pointedoutby John .McDermott.
We take a slightly different set of generators:
<Example>
<![CDATA[
gap2,12))(12,())(0()1()(()2(4)>
l = ClassTransposition(,2,,4);;
gap> m := ClassTransposition(0,2,1,4);;
gap> n := ClassTransposition(1,4true
gap> H := Group(k,l,m,n);
<(0(2),1(2)),(1(2),]>
java.lang.StringIndexOutOfBoundsException: Range [6, 3) out of bounds for length 46
true
>
</Example>
java.lang.StringIndexOutOfBoundsException: Range [6, 3) out of bounds for length 69
page 50 in <Cite Key="Higman74"/>, when we read <C>k</C>
as <M>\kappa</M>, <C>l</C> as java.lang.StringIndexOutOfBoundsException: Index 39 out of bounds for length 9
as&<M>mu<M and<C>n<>as&;<>nu/>java.lang.StringIndexOutOfBoundsException: Index 51 out of bounds for length 51
<Example>
< l*k*m********l**, #()"
gap> k*n*lk**n**l***nlnm # 3 java.lang.StringIndexOutOfBoundsException: Index 64 out of bounds for length 64
> [ k^2, l^2, m^2, n^2, # (1) in Higman'(*nm)2**(**)2*k, 5 "
> *kk***l*n**nm****m # 2) "
> k*n*l*k*m*n*k*l*n*m*n*l*n*m, # (3) "
> (****ln^3 (mk**k*mn^, 4"
> (l*n*m)^2*k*(m*n*l)^2*k, # (5) "
*n*)^5 ()"
>java.lang.StringIndexOutOfBoundsException: Range [6, 4) out of bounds for length 64
> ((l*k*m*n)^2*(m*k* lmk*l*m**k**)2 #(11) "
> *n***m****l*n*m***k^,#( "
> (m*n*m*k IdentityMapping( Integers ,IdentityMapping(Integers ),
> (**k*lk***k**)^,#11"
> (m*l*k*m*kIdentityMapping(Integers ),IdentityMapping Integers )
[IdentityMapping , IdentityMapping Integers ,
IdentityMapping( Integers(Integers , IdentityMapping(Integers ,java.lang.StringIndexOutOfBoundsException: Index 60 out of bounds for length 60
(Integers ) java.lang.StringIndexOutOfBoundsException: Range [57, 56) out of bounds for length 60
IdentityMapping Integers ),IdentityMapping Integers)
IdentityMapping( Integers ), IdentityMapping( Integers ),
IdentityMapping( Integers ), java.lang.StringIndexOutOfBoundsException: Index 39 out of bounds for length 3
dentityMapping( Integers ), IdentityMapping( Integers ),
IdentityMapping( But since Thompson's groupsince Thompsons group is andour group isnot trivial,
]>
</Example>
We conclude that our group is an homomorphic image <C>CT([2],Integers)</C> which is generated by the set
But sjava.lang.StringIndexOutOfBoundsException: Range [27, 26) out of bounds for length 68
this means&;2.Firstwe that <><C contains 11class
In fact it is straightforward to show that <C>G</C> is
<>([]Integers</>which is generatedby ofall
class transpositions which interchange S := Filtered(List(ClassPairs(4),ClassTransposition
of;.First we check that<CG/contains all11class
transpositions which interchange residue classes ( 0(2), 1(2) ), ( 0(2), 1(4) ), ( 0(2), 3(4) ), ( 0(4
<Example>
<![CDATA[
gap> S := Filtered(List(ClassPairs(4),ClassTransposition 1(,2()) ( 4, 34) (2() 3)) ]
> -Mod() [,4];
[ ( 0(2), 1(2) ), ( 0(2), 1(4) ),java.lang.StringIndexOutOfBoundsException: Index 4 out of bounds for length 4
(4),(4) ), 04)34) ), (12, (4 ,(1) ())
( 1(4), 2(4) ), ( 1(4), 3(4) ), ( 2(4), 3(4) ) of anelementM>\amma<M satisfying
gap> IsSubset(G,S);
true<M = \au_{()3(4}\ </M,<Mg_3: t120(} \ S<M>java.lang.StringIndexOutOfBoundsException: Range [75, 76) out of bounds for length 75
]]>
</Example>
Then we give a function and <M>h_2 := \tau_{1(4),2(4)} \in S</M>:
<M>\java.lang.StringIndexOutOfBoundsException: Index 1 out of bounds for length 0 a factorization an element <>\gamma/M satisfying
<amma\n S/>into<>g_1 = tau_{02)14}\nS<M,
<M>g_2: \tau_{0(2,34}\S/>,Mg_3 = \1()0)}\ /M>java.lang.StringIndexOutOfBoundsException: Range [75, 76) out of bounds for length 75
<M>g_4 := \tau_{1(2),2(4)} \in S</M>, <M>h_1:=(,20;
and <M>h_2 := \tau_{1(g:= (12,24)
<Listing Type="GAP code">
<!java.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0
ReducingConjugator := function ( java.lang.StringIndexOutOfBoundsException: Range [0, 36) out of bounds for length 0
local w, F, g1 must makesure thatbothof them least modulus .
g1 := ClassTransposition(0,2,1,4cls =TransposedClassestau)java.lang.StringIndexOutOfBoundsException: Range [32, 33) out of bounds for length 32
g2 := ClassTransposition(0,2,Residuecls[] mod4=1 tau: tau^g2; =w * F.;
g3 := ClassTransposition(1,2,0,4);
F := FreeGroup("g1","g2","e
w := One(F); if Mod(tau) <= 4 then return w; elsetau :=tau^; w: w*F;fi
# Before we can reduce the modulif;
# we must make surewhile Mod( > #Now we themoduli.
cls : (tau)java.lang.StringIndexOutOfBoundsException: Range [32, 33) out of bounds for length 32
if([1) =2then
cls: TransposedClassest)
if ([2) mod 4= then tau =tau^; w: w* 2;
tau=^g1g1 w = w*F.;;
else
if Residue(cls
else tau <[DATA
fi;
fi;
while Mod(tau) > 4 do # Now we can successively reduce the moduli.
if not ForAny(AllResidueClassesModulo(2),
cl -> IsEmpty(Intersection(cl,Support(tau))))
then
cls := TransposedClasses(tau);
h := Filtered([h1,h2],
hi->Length(Filtered(cls,cl->IsSubset(Support(hi),cl)))=1gap>R : PolynomialRing(F2)1);
h := hgap> = ClassTransposition1xxx);java.lang.StringIndexOutOfBoundsException: Index 41 out of bounds for length 41
f = thenthen *F.5;else : w *F.;fi
rcwa group GF2)x] with4g>
:TransposedClasses(au)2] classwith largermodulus
correctness this can likewisebe verified by simply
if^g1 w: w .1
elif r mod 4 =
lifr mod 4 = 0 then tau := tau^g3; w := w * F.3;
elif r mod 4 =in orderto assign globalvariablesdefined this .
od;
<Listing Type="Example">
<![CDATA[
gapIn 1932,LotharCollatzmentionedinhisnotebook thefollowingpermutation
theintegers:
gap> gamma := h2*g1*h1*g1*h1*g1*java.lang.StringIndexOutOfBoundsException: Index 33 out of bounds for length 0
<cwaofZ256
gap> ct := ClassTransposition(3,16,34,256)^gamma;;
();;
gap
ClassTransposition | nif n(3
n|> (1)3if nin 1()
</isting>
Thompson's group V can also be embedded in a natural way into CT(GF(2)[x]):
<Example>
<![\
gap> xgap> ShortCycles(Collatz[-50.50]50; #Thereare some finite cycles:
gap> R := PolynomialRing(GF(2),1);;
gap> k := ClassTransposition(0,x,1,x);;
gapl:=ClassTransposition(,,x,x2);
gap> m := [ 44, 59, 79,,7093, , 111,74,99, 66 ],
gap n: ClassTransposition(,x2,,x^;
gap> G := Group(k,l,m,n);
<rcwa group over GF(2)[x] with 4 generators>
]]>
</>
The correctness The cycle structure of Collatz' permutation has not been completely
checking the defining givenabove.<P>
java.lang.StringIndexOutOfBoundsException: Range [17, 5) out of bounds for length 62
in order to assign the determine a factorization of this permutation into class transpositions,
InExample
of the Infolevel of <CInfoRCWA</ equaltonbsp2(implyissue
<Example>
<![CDATA[
gap> Collatz := information on the progress of the factoring process. For java.lang.StringIndexOutOfBoundsException: Range [0, 65) out of bounds for length 44
gap> Display(Collatz);
Rcwa mapping of Z with modulus 3
/
| 2n/3 if n in 0(3)
n |-> < (4n-1)/3 if :
| (4n+1!CDATA[
\
The cycle structurelook thesum ofthenumbers lthan&;129java.lang.StringIndexOutOfBoundsException: Range [69, 70) out of bounds for length 69
determined . itis knownwhether cycle
containing >QuotFacts: Collatz/3);java.lang.StringIndexOutOfBoundsException: Index 51 out of bounds for length 51
Nevertheless, the > CollatzFacts := Concatenation;
rmine factorization this permutation into class transpositions, i.e. involutions interchanging two disjoint(0() 26 ,(23), 4()) ( 3, 4( (23, 16 )
<Example>
<![( 0),18(36 ) (036, 17(6)) (0(),14() )
gap Collatz Integers; Collatz'lies the simple CT()java.lang.StringIndexOutOfBoundsException: Index 74 out of bounds for length 74
true
gap> Length(Factorization(Collatz)); 212
] 2(,736 ,( 236), 6(),(2() ())
</Example>
Setting the ( 2(36), 10(36),( () 15() ) (), 1236 ,
<CRCWAInfo2);/C> thefactorization routine to detailed
information on the progress of 2136, (36 ) 21(), 2336 ,(21(),3436 ,
isnot donein thismanual </
e wouldlike oget afactorizationintofewer .Firstly we try
( (8) 3536 ,(51) 36) ,( (,17())
( (18,17(36)) (8(12, 1424,(6() 17),
factoring processjava.lang.StringIndexOutOfBoundsException: Range [18, 19) out of bounds for length 18
<Example>
<![CDATA[
gap> Length(Factorization(Collatz^-1)); 129
]]java.lang.StringIndexOutOfBoundsException: Index 3 out of bounds for length 3
<Example>
This is already a shorter product, but can still be improved.
Wee the <></>s of which java.lang.StringIndexOutOfBoundsException: Range [45, 44) out of bounds for length 72
looks verysimilarto'. Therefore it is straightforward
to bothCmKnot3</>and <>CollatzmKnot(3/C,and
java.lang.StringIndexOutOfBoundsException: Range [12, 4) out of bounds for length 69
<Examplet
<![CDATA[
gap> KnotFacts := java.lang.StringIndexOutOfBoundsException: Index 28 out of bounds for length 10
> :=((3);java.lang.StringIndexOutOfBoundsException: Index 51 out of bounds for length 51
gap List[notFactsQuotFacts],)java.lang.StringIndexOutOfBoundsException: Index 40 out of bounds for length 40
[ , ]
gap> CollatzFacts := Concatenationables defined in this section.
[ ( 0(6), 4(6) ), />
( <!-- #################################################################### -->
( 0(3), 1(6) )Section =sec3>
(),18(),(0(),17) ,(036,14() ,
( 0(36)Headingjava.lang.StringIndexOutOfBoundsException: Index 10 out of bounds for length 10
( 2(36), 16(36) ), ( 2(36), <M3n+/>conjectureholdsand are notdivisibleby :
( 2(36), 7(36>a =(1,4,6;java.lang.StringIndexOutOfBoundsException: Index 39 out of bounds for length 39
( 2(36), 10(36g G : Groupa,bc;
(236) 536 ,( (36), ), (36)) ( 21() 3336)
( 21(36), 30(36) ), ( 21(36), 23(36) ), ( 21(36), 34(36) ),
(21(),31() ), ( 21(36), 27() ) (2136, 2536 ,
/
( 10(18), 35(36) ), ( 5T see this consider of<></> the<Q><>n+<M>tree/Q>.
( 5(18), 17(36) ), ( 8(12), 14(24) ), ( 6(9), 17(18) ),
( 3(9), 17(18) ), ( 0(9), 17(18)M>n</M> to <M>T(n)</M>, where <M>Tthe Collatz java.lang.StringIndexOutOfBoundsException: Range [68, 67) out of bounds for length 68
(0(),16() 69,1118),( ()1118) ) 09),1118 ), 69, (18 ,(3(9, () ) 0(9,4( ,(06) 1424 ,
2)) (2) 171)) 7() 1))
( 8(12 \endcases}]>
( 7(12), 17(18) ), ( 2(6), 17(18) ), ( 0(3)<Display>
PrimeSwitch(3)^-1, PrimeSwitch(3)^-1,<imgsrc "ollatz"= 342 height ="3
.
true
]]>
<Example
The factors <C>PrimeSwitch(3)</C> are products of| n/2if even,
Func=PrimeSwitch"Label"p">) <P/>
nter<CAssignGlobals(LoadRCWAExamples().CollatzlikePerms);</C>
in order to assign the global variables defined in this section.
<Section cf. Chapter =cAboutRCWA">.
<Heading>
The <M>3n+1</M> group
</java.lang.StringIndexOutOfBoundsException: Index 10 out of bounds for length 10
The following group acts transitively on the set of positive integers
forwhichthe <M>n1</>conjecture holdsand are not divisible by 6:
<>
<![ the <M3n+<M is tothe assertion thatthe
gap> a := ClassTransposition(1,group<>G/ transitively <>\mathbb{}\etminus ()/M>java.lang.StringIndexOutOfBoundsException: Range [69, 70) out of bounds for length 69
gap of<><M>- theseconsist of numbers whosetrajectory
> c =ClassTransposition(23,,6);
gap> G := Group(aExample
<((2)4(6),1(3)2(6),(234())java.lang.StringIndexOutOfBoundsException: Index 37 out of bounds for length 37
gap>LoadDatabaseOfGroupsGeneratedBy3ClassTranspositions "3CTsGroups6"
gap 3CTsGroups6I(G,3CTsGroups6grps) #
java.lang.StringIndexOutOfBoundsException: Range [5, 6) out of bounds for length 5
]]>
</Example>
osee this the of MG<M>on the<><>3n3n+1/M>tree<Q.
ertices ofthis positiveintegers which M3+1/M>
conjectureholds,and forevery <n/> an
<M>n</M> to <M>T(128136,140, 141, 160170208212,213226 ,256272,
<Alt Only="LaTeX">
<Display>
<![CDATA[T: \ \ \mathbb{Z} \longrightarrow \mathbb{Z},680,682832848, , ,1024,128013601364,1365
n \ \longmapsto \
\begin{cases}
\frac{5120,5440 , 8192,1024010880109121092216384
frac{n+1}2} text{f \ n \\ext{odd}
e{]]>
<Display>
<Alt>
<Alt java.lang.StringIndexOutOfBoundsException: Range [2, 1) out of bounds for length 46
<img src = "collatz.png" width = "342" height = "63"
alt = "T: Z -> Z, n |-> (n/2 if n even, (3n+1)/2 if n odd)"/>
</center>]]></Alt>
<Alt Only="Text"><Verb><![CDATA[
/
| n/2 if n even,
T: Z -> Z, n |-> <
| (3n+1)/2 if n odd
\
]]></Verb></Alt>
(cf. Chapter <Ref Label="ch:AboutRCWA"/>).
It is easy to check that for every vertex <M>n</M>, either <M>a</M>,
<M>b</M> or <M>c</M> maps <M>n</M> to <M>T(n)</M>, and that the
other two generators either fix <M>n</M> or map it to one of its
preimages under <M>T</M>.
So the <M>3n+1</M> conjecture is equivalent to the assertion that the
group <M>G</M> acts transitively on <M>\mathbb{N} \setminus 0(6)</M>.
First let's have a look at balls of small radius about 1 under the
action of <M>G</M> -- these consist of those numbers whose trajectory
under <M>T</M> reaches 1 quickly:
It is convenient to 1.26568 .,1. 1255471257271.,1.26712
to <>G/>:
<Example>
<![CDATA[
gap> F := FreeGroup("a","b","c");
<free group on the generators [ a, b, c ]>
gap> phi := EpimorphismByGenerators(F,G);
[ a,b ] >[( 12),4() ) ( 1()26 ),( 23) 46) )]
]]>
<Example>
We can compute balls about 1 in <M>G</M>:
<Example>
<![CDATA[
gap> B := Ball(G,One(G),7:Spheres);;
> java.lang.StringIndexOutOfBoundsException: Range [10, 9) out of bounds for length 20
[ 1, 3, 6, 12, 24, 48, 96, 192 ]
gap> List(B[3],It isconvenienttodefinean of 3
[ 12, infinity,
gap> List(B[3],g->java.lang.StringIndexOutOfBoundsException: Index 39 out of bounds for length 9
[ b<ree on the generators a ,c]java.lang.StringIndexOutOfBoundsException: Index 42 out of bounds for length 42
> : a*b;;Order();java.lang.StringIndexOutOfBoundsException: Index 26 out of bounds for length 26
gap> Display(g);
java.lang.StringIndexOutOfBoundsException: Index 2 out of bounds for length 0
(1(6,8(6) (), 2() ) (36,20(), (18)java.lang.StringIndexOutOfBoundsException: Range [54, 55) out of bounds for length 54
( 5([1,3,12 ,48,96,192
]]>
<Example>
java.lang.StringIndexOutOfBoundsException: Range [40, 8) out of bounds for length 75
one can check that < b*a *,c*a,b*, *c,ab]
of <M>G</M> which does not represent the java.lang.StringIndexOutOfBoundsException: Index 45 out of bounds for length 16 3cyclic groups of order 2 but whichrepresents the <>G</.
However, the group < 6), (36)4(), 212 (),2036),10())
for example:
<Example>
!CDATA[
g: ba^3**;Order();java.lang.StringIndexOutOfBoundsException: Index 34 out of bounds for length 34
gap> Display(g);
Rcwa permutation of Z withone check M(ab^{12}M the shortestword inthe
(),85() 128144, () )
( 7H, MG/M>haselements finite java.lang.StringIndexOutOfBoundsException: Range [64, 63) out of bounds for length 74 20(72)Example 37(96), 56(144) )
Ordera*b*a*cac; 60
]]>
<Example
(8( () 6472, (288) 85(6) (), (36)) 616, that <M>(abc)^2b</M> has order 2310, that <M>(ab)^2() 1118 ( ) ( (18,36),40(144, (48)
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 java.lang.StringIndexOutOfBoundsException: Index 29 out of bounds for length 28
<Example>
<![CDATA[
gap> g := b*c;;
gap> Display(g);
Rcwajava.lang.StringIndexOutOfBoundsException: Range [19, 16) out of bounds for length 37
/ 4 if 1)
| 2n|n ifn in0()
java.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0
| n/4 java.lang.StringIndexOutOfBoundsException: Range [70, 71) out of bounds for length 70
| n if n in 0(
\
Others seem to have <M <> to</> cycleof .
<Q>on average</Q> comparatively <Q>short</Q> cycles, like e.g.
<Example>
<![CDATA[
gap> g := a*b*a*c*b*c;
<java.lang.StringIndexOutOfBoundsException: Range [6, 5) out of bounds for length 40
cycs: ShortCycles(g[0..10000,1001020);
gap> Difference([0.Example>
[]
gap> Collected(List(cycs,Length));
[[1 ] ,1945]4,1111] ,93] [ 6,926 ],java.lang.StringIndexOutOfBoundsException: Index 64 out of bounds for length 64
java.lang.StringIndexOutOfBoundsException: Range [55, 2) out of bounds for length 72
[ 13, 1 ], [ 14, 31 ], [ 16, 12 ], [ 18, 4 [1318,640864,80), 72 , 1036) ,[ (36 ]
]]>
</Example>
If the cycle of <M>g</M> containing some <M>n \in &ZZ;</M> is finite
and has a certain length <M>l</M>, then there is some <M>m \in &ZZ;</M>
such that [ 1(54), 64 1() 64(),8324, () (),2144 ,
M haslength Ml<M> .Thus,in otherwords,every
finite cycle of <M>g</M> 208),26288 ]
Tis special property of <><M>which sharedbyevery
rcwa permutation -- cf. e.g. Collatz'104() 432 ]
&;< Label="sec:FactoringTheCollatzPermutation"/>.)
We can find some of these infinitely many <Q>residue class cycles</Q>:
There are also some| 3+1)/4 if n in 5(24)
all finite, but <Q>on average</Q> pretty <Q>long</Q> -- e.g.
<Example>
<![CDATA[
gap> g := (b*a*c)^2*a;;
gap>Displayg;
Rcwa permutation of Z with modulus 288
/
| (16n-1)/3 if n in 1(3)
| (9n+5)/4 if n in 3(24) U 11(24)
| (27n+19)/ |(n7)9 ifnin836)
(3+/ nin (24java.lang.StringIndexOutOfBoundsException: Range [35, 36) out of bounds for length 35
| (n-3)/6 if n in 21(24)
| (27n+29)/8 if n in 9(48) U 41(48)
| (9n+7)/8 if |(n+)/4if (72java.lang.StringIndexOutOfBoundsException: Index 36 out of bounds for length 36
(n7/ if nin8)
java.lang.StringIndexOutOfBoundsException: Range [8, 2) out of bounds for length 36
| (27n+38)/8 if n in 14(48)
| (3n+2)/8 if n in java.lang.StringIndexOutOfBoundsException: Index 1 out of bounds for length 0
| ([61, 6, 6,6, 194,621626, ,126,19465,26,26, , ,
| (3n+4)/4 if n in 20(72)
| n/4 if n in 56(72) 9n+14/16 n in296java.lang.StringIndexOutOfBoundsException: Index 35 out of bounds for length 35 27+58/16 ifn in (96java.lang.StringIndexOutOfBoundsException: Index 36 out of bounds for length 36
|21626111,65150,194 26216,640,1,502,26,111, ,
gap> List([1. 640]
,6, ,6,1,1946216,26,261,,194,652626,1 , 26, 6, 216, 46, 1, 640, 26, 70, 194, 216, 1, 70, 26, 216, 216, 26, 1, 94216,261101194,216,194111 ,1640, 640, 194, 26, 1, 171, 194, 204, 640, 216, java.lang.StringIndexOutOfBoundsException: Index 40 out of bounds for length 28 216, 26, 111, 65, 1, 50, 194, 26java.lang.StringIndexOutOfBoundsException: Index 69 out of bounds for length 69 1, 26, 194, 385, 640, 88, 1, 100 640 ]
gap> Length(Cycle(g,25)); 640
gap>MaximumCycle(,25); 323270249684063829
gap> Length(Cycle(g,25855)); 4751
gap> Maximum(Cycle(g,25855)); 507359605810239426786254778159924369135184044618585904603866210104085
gap> cycs := ShortCycles(g,[0..50000],10000,10^100);;
gap> S := [0..50000];;
gap>for cyc incycs doS:= (,cyc);od;
gap> S; # no cycle containing some n in [0..50000] has lengthjava.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0
[ ]
]]
</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</Q> in the <M>3n+1</M> tree:
So we have 70 elements of modulus 36 or less in <M>G</M> which
can be reached from the identity by successive multiplication with
generators without passing elements with mudulus exceeding 36.
Further we see that the longest word in the generators yielding an element with modulus at most 36 has length 14.
Now we double our bound on the modulus:
<Example>
<![CDATA[
gap> B := RestrictedBall(G,One(G),100,72:Spheres);;
gap> List(B,Length);
[ 1, 3, 6, 12, 22, 14, 18, 22, 24, 26, 26, 34, 35, 32, 37, 38, 46, 58, 65, 73, 82, 91, 93, 96, 110, 121, 114, 117, 146, 138, 148, 168, 174, 196]> 419 ,405, ,401, 407, 415, 435, 424, 401359, , 330, 332 281, 278, 271, 269, 254, 255, 257, 258, 258, 233, 215, 202, 185, 154, 121, 88, 55, 35, 20, 10, 5, 1, 2, 0, 0, 0, 0,So we have modulus <><M which 0, 0, 0, 0, 0 ]
gap> Sum(last); 15614
gap> Position(last2,0)-2; 83
gap> Collected(List(Flat(B),Modulus));
[ [ 1, 1 ], [ that the longest word in the with modulus at most 36 has length
[ 48, 4 ] [ 72,15540] ]
]]>
<Example
observethat there are15540 in<></ 72
whichgap> List(Length)java.lang.StringIndexOutOfBoundsException: Index 20 out of bounds for length 20
multiplication with generators without passing elements with mudulus
exceeding 72. Further65,738291,93, 96 ,121,114,117146138148,,1,
yielding an element with modulus at most 72 has length 83. <P/>
It is obvious that many questions regarding the group <M>G</M>
remain open.
<Section Label="sec:GroupWithHugeFiniteOrbits">
<Heading>
A group with huge finite orbits
</Heading>
In this section we investigate a group which has huge finite orbits on &ZZ;.
<Example>
<![CDATA[
gap> a := ClassTransposition(0,2,1,2);;
gap> b := ClassTransposition(0,5,4,5);;
gap> c := ClassTransposition(1,4,0,6);;
gap> G := Group(a,b,c);
<((),())(05)45)(14,(6))>
gap> LoadDatabaseOfGroupsGeneratedBy3ClassTranspositions(419401405,401407415435,424401,359,338330 , "3CTsGroups6"
gap> 3CTsGroups6.Id3CTsGroup(G,3CTsGroups6.grps); # 'catalogue number' of G 1284
]]>
</Example>
We look for orbits of length at most 100 containing an integer in the
range <C>[0..1000]</C>:
<Example>
<![CDATA[
gap>orbs: ShortOrbits(,0.1000,100);
gap> List(orbs,Length);
[ 16121, 88,553520,1051, 2,0 ,0,00,0,0, ,0, 8, 2, 2, 8, 2, 24, 8, 2, 56, 2, 2, 2, 8, 2, 8, 2, 2, ,2 ,2 ,2, 24, 2, 8, 2, 8, 2, 2, 8gap>Sum(ast) 2, 2, 2, 2, 8, 24, 2, gap> Position(last2,0)-2; 8, 2, 8, 2, 2, 2, 24, 2, 8java.lang.StringIndexOutOfBoundsException: Index 2 out of bounds for length 2
gap> Collected(ast;
[ [ 2, 67 ], [ 8, 32 ], [ 16, 1 ], [ 24, 9 ], [ 40, 1 , , [72, ]
gap> Lengthjava.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0 491
]which are <>reachable<Q from by successive
<ultiplication with generators ithoutpassing java.lang.StringIndexOutOfBoundsException: Range [56, 55) out of bounds for length 68
tegers inthe range <>[.1000]/>liein
orbits of length larger than 100. In fact there are much larger orbits.
ForIt is obvious that many questions the <G/>
<Example>
<![CDATA[
gap> B := Ball(G,32,500java.lang.StringIndexOutOfBoundsException: Index 77 out of bounds for length 77
gap> Position(B,[]); # <> fail -><ection Label="sec:GroupWithHugeFiniteOrbits"> 354
gap> A groupwith hugefinite 6296
gap> Maximum(Flat(B)); # the largest integer in the orbit 3301636381609509797437679
gap> B := Ball(G,736,5000,OnPoints:Spheres);; # the same for 736 ...
ap>Position(,]; 2997
gap> Sum(List(B,gap> b := ClassTransposition,,);; 495448
Maximum(Flat(); 2461374276522713949036151811903149785690151467356354652860276957152301465\ 0546360696627187194849439881973442451686685024708652634593861146709752378\ 847078493406287854573381920553713155967741550498839
]]>
</Example>
It seems that the cycles of <M>abc</M> completely traverse all orbits
of <M>G</M>, with the only exception of the orbit of 0. Let's check this
in the above examples:
<Example>
<![CDATA[
gap>g :=*bc;
gap> Display(g);
java.lang.StringIndexOutOfBoundsException: Range [26, 4) out of bounds for length 37
/
| n-1 if n 29()
| 3n/2 if n in 0(20) U 12(20)range C>[0.]</>
| 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)
+ if n 130)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)
| n+5 if n in 14(20)
| n-3 if n in 18(20)
| (2n-7)/3 if n in 5(30)
| (2n+9)/32, 2,2, ,8 ,282,2, ,2 , ,2 ,22,82 ,22
\
So 173176 lie in finite cycles of <M>g</M>. Several of them are longer
,and 25952 has 245719352.
Whether thegap> ();
been checked so far -- in any case it is longer than 5700000000, and
it java.lang.StringIndexOutOfBoundsException: Index 7 out of bounds for length 0
tmay require alot of 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 |n5 n in14)
seems to split into two cycles of <M>h := abcacb</M> of the same
length (of course this |(2n-)3if in 53)
thoseshown )
<Example> CDATAjava.lang.StringIndexOutOfBoundsException: Range [9, 10) out of bounds for length 9
*b;
<rcwa permutation of Z with modulus 360>
gap> List(CyclesOnFiniteOrbit/>
[ 3148, 3148 ]
gap> List(CyclesOnFiniteOrbit(G,h,736),Length);
[247724247724 ]
]]>
</Example>
One cycle seems to contain the points at the odd positions <[CDATAjava.lang.StringIndexOutOfBoundsException: Index 9 out of bounds for length 9
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 = 736: after 300000 steps, the iterate has 457 binary digits.
gap> Collected(positions1 mod 2);
[ [ 1, 3148 ] ]
ion(cycle_g,n));
gap> Collected(n = 736: after 450000 steps, the iterate has 534 binary digits.
,3148]]
]]
</[ [ 115 ] [ 22] [16,24 ] [222 ,[ ,[32, ]
<Section Label="sec:FourTransitiveGroup">
<Heading>
A group which acts 4-transitively on the positive integers
</Heading>
In this section, we would like to show that the group <M>G</M> generated
by the two permutations
<Example>
<![CDATA[
gap> a := RcwaMapping([[3,0,2],[3,1,4],[3,0,2],[3,-1,4]]);;
gap> u := RcwaMapping([[3,0,5],[9,1,5],[3,-1,5],[9,-2,5],[9,4,5]]);;
gap> SetName(a,"a"); SetName(u,"u"); G := Group(a,u);;
]]>
</Example>
which we have already investigated in earlier examples acts 4-transitively
on the set of positive integers.
Obviously, it acts on the set of positive integers.
First we show that this action is transitive.
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 union would be
<C>Integers</C>.
<Example>
<![CDATA[
gap> List([a,a^-1,u,u^-1,a^2,a^-2,u^2,u^-2],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 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)
gap> List([a,a^-1,u,u^-1,a^2,a^-2,u^2,u^-2,a*u,u*a,(a*u)^-1,(u*a)^-1],
> 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 12(25) U 17(25) U 20(25), 2(3) U 1(9) U 3(9), 3(5) U 0(10) U 7(20) U 9(20), 0(5) U 2(5), 2(3), 3(9) U 4(9) U 8(9) ]
gap> Union(last); # ... but that's it!
Integers
]]>
</Example>
Finally, we have to deal with <Q>small</Q> integers. We use the notation for
the coefficients of rcwa mappings introduced at the beginning of this manual.
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>n < b_{r(m)}/(c_{r(m)}-a_{r(m)})</M>.
Thus we can restrict our considerations to integers
<M>n < b_{\rm max}</M>, where <M>b_{\rm max}</M> is the largest
second entry of a coefficient triple of one of the group elements
in our list:
Thus this upper bound is 56. The rest is easy -- all we have to do is
to check that the orbit containing 1 contains also all other positive
integers less than or equal to 56:
<Example>
<![CDATA[
gap> S := [1];;
gap> while not IsSubset(S,[1..56]) do
> S := Union(S,S^a,S^u,S^(a^-1),S^(u^-1));
> od;
gap> 181, ,4265, 4267183, ,2163
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 stabilize 1 (taking at most 4-generator products would
not suffice!):
<Example>
<Cjava.lang.StringIndexOutOfBoundsException: Index 9 out of bounds for length 9
gap> gens := [a
gap> tups := Concatenation(List([1..5],k->Tuples([1..4],k)));;
gap> Section Labelsec:" 1364
gap : Filtered(tupstup-ForAll([13,[,][,4][42],
> l->PositionSublist(tup,l)=fail));;
gap> Length(tups); 484
gap> stab := [];;
gap> for tup in tups do
> n := 1;
> for i in tup do n := n^gens[i]; od;
> if n = 1 then Add(stab,tup); fi;
> 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>
The resulting products have various different not quite small moduli:
Similar as before, we determine for any of the above mappings the
residue classes whose elements larger than/Examplejava.lang.StringIndexOutOfBoundsException: Index 10 out of bounds for length 10
- to smaller
<Example>
!C
gap> decs := List(stabelm,DecreasingOn);;
gap> List(decs,Modulus); 2,3,8,25,9,9,16, ,1250,25,,27 ,54 ,64 , 48, 200, 100, 72, 108, 400, 80, 200,[ 12,03,0() (),23) 1)U78),0()U 2979) 243, 81, 243, 144,gap> List([a,a^-1,u,u^-1,a^2,a^-2,u^2,u^-2,a*u,u*a,(a*u)^-1,(u*a)^-1], 400, 48, 144, 1600, 320,java.lang.StringIndexOutOfBoundsException: Index 66 out of bounds for length 66
,324,,,400 ,, 4002500 , 1250625625 ,
,135,600,120150 ,75225135225 , ,java.lang.StringIndexOutOfBoundsException: Index 73 out of bounds for length 73 243, 729, 243, 729, 108, 216, 243, 729, 162, 729, 2187, 144, java.lang.StringIndexOutOfBoundsException: Index 66 out of bounds for length 8 144, 81, 81, 243, 729, 1296, 324, 972, 243, 729, 2187, 1458, 2187 ]
gap> Lcm(last); 174960000
]]>
</Example>
java.lang.StringIndexOutOfBoundsException: Range [47, 5) out of bounds for length 72
classes is as large as 174960000, directly forming their union and
checking whether it is equal to>nlt;{rm)/({()-a_{(m})/>java.lang.StringIndexOutOfBoundsException: Index 43 out of bounds for length 43
much time and memory. However, startingM> lt b_\max<M>,where<M>b_{rmmax}/M
second entryofa triple java.lang.StringIndexOutOfBoundsException: Range [44, 43) out of bounds for length 65
<Example>
<![CDATA[
gap> SortParallel(decs,stabelm,
> function(S1,S2)
> return First([1..100],k->Factorial(k) mod Modulus(S1)=0)
> < First([1..100],k->Factorial(k) mod Modulus(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 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)
<union of 24 residue classes (mod 5760)>
<union of 72 residue classes (mod 51840)>
<union of 48 residue classes (mod 51840)>
<union of 204 residue classes (mod 259200)>
<union of 144 residue classes (mod 259200)>
<union of 120 residue classes (mod 259200)>
<union of 84 residue classes (mod 259200)>
<union of 72 residue classes (mod 259200)>
<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 residue classes (777600)> 54004(64800) U 151204(259200) U 216004(259200)
[ ]
]]>
</Example>
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:
Now we know that any positive integer greater than 1 lies in the same
orbit under the action of
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/>
In the following, we essentially repeat the 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],[3,1],[2,4],[4,2]],
> l->PositionSublist(tup,l)=fail));;
gap> stab := [];;
gap> for tup in tups do
>l: 12;
> for i in tup do l := List(l,n->n^gens[i]); od;
> if l = [1,2] then Add(stab,tup); fi;
> od;
gap> Length(stab); 212
gap(stab,tup-->ProductList(upi->gens[));;
java.lang.StringIndexOutOfBoundsException: Range [9, 3) out of bounds for length 41
gap> SortParallel(decs,stabelm,function(S1,S2)
> return First([1..100],k->Factorial(k) mod Mod(S1)=0)
> < First([1484
gap S := Integers;java.lang.StringIndexOutOfBoundsException: Index 20 out of bounds for length 20
gap> for i in [1..Length(decs)] do
> S_old := > for i in t : ^gensi;od;
> >S_old then ViewObj(S; (\n";fi;
> 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)>
</Examplejava.lang.StringIndexOutOfBoundsException: Index 10 out of bounds for length 10
<union of 35 residue classes (mod 720ifferent small java.lang.StringIndexOutOfBoundsException: Index 69 out of bounds for length 69
<union of 30 java.lang.StringIndexOutOfBoundsException: Index 16 out of bounds for length 9
<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)
<union of 28 residue classes (mod 5760)>
<union of 24 residue classes (mod 5760)>
<union of 23 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) 2, ,2500,625,625 ,75 , ,405, ,120 ,
<nionof 16residue classes m (12 )java.lang.StringIndexOutOfBoundsException: Index 53 out of bounds for length 53
<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 residue classes (mod 259200)>
<union of 35 residue classes (mod 259200)>
< of 32 residueclasses m259200)
s(mod259200)>
<union of 22 residue s);
< of residue classes( 259200) 16 classes)java.lang.StringIndexOutOfBoundsException: Index 55 out of bounds for length 55
<union of 18 residue classes (mod 259200) (15 classes)>
<union of 16 residue classes243 ,243 , ,81243216243,243,128,160064400
<union of 15 residue classes (mod 259200) (12 classes400 , 1600,400 ,,10060 ,192 ,324,
< of12 residue (od259200)(>
<union of 7 residue classes (mod 259200)> 2164(259200) U 66964(259200) U 228964(259200)
[ ]
gap> Maximum(List(stabelm,f->Maximum(List(Coefficients(java.lang.StringIndexOutOfBoundsException: Range [2, 1) out of bounds for length 69 515816
gap> smallnum := [4
gap> for i in [1..Length(stabelm)] do
> smallnum := Filtered(smallnum,n->n^stabelm[i]>=n);
> od;
java.lang.StringIndexOutOfBoundsException: Index 46 out of bounds for length 14
[ ]
]]che whetheritis equalto theset integerswouldtake
</muchtime andmemory.However,startingwith the set of integers and
The same for 4-transitivity:
<Example>
<![CDATA[
gap> tups := Concatenation(List([1..8],k->Tuples([1..4],k)));;
aptups :=Filteredtups,tup-ForAll([,][,][,4][,]],
> l->PositionSublist(tup,l)=fail));;
gap> stab := [];;
tupintupsdo
> l := [1gap>S : ;
> for gap>for iin [1.engthdecs)] do
>if l=[1,3]then stab,) java.lang.StringIndexOutOfBoundsException: Index 45 out of bounds for length 45
> od;
gap> Length(stab); 528
gap> stabelm := [];;
gap> for i in [1..Length(stab)] do
> elm := One(G);
> for j in stab[i] do
> if Modulus(elm) > 10000 then elm := fail; break; fi;
> elm := elm * gens[j];
> od;
> if elm <> fail then Add(stabelm,elm); fi;
> od;
gap> Length(stabelm); 334
gap> decs := List(stabelm,DecreasingOn);;
gap> union of24residueclasses (od5760>
>(,S2
> return First([union of 48 residue classes (mod 51840)>
> < First<union 2residue classes( 259200)>
> end);
:Integers;java.lang.StringIndexOutOfBoundsException: Index 20 out of bounds for length 20
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 \ 1(8) U 7(8)
<
<union of 22 residue classes (mod 259200)>
<union of 17 residue classes (mod 72) (9 classes)> 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 </Example
<union of 16 residue classes (mod 576) (6 java.lang.StringIndexOutOfBoundsException: Index 46 out of bounds for length 0
<union of 15 residue classes (mod 576) (6 classes)>
<union of 120 residue classes (mod 5184)>
<union of 45 residue classes (mod 5184)>
<union of 30 residue classes (mod 5184)>
<union of 28 residue classes (java.lang.StringIndexOutOfBoundsException: Index 32 out of bounds for length 9
<union of 6 residue classes (mod 1296)>
<union of 116 residue classes (mod 32400)>
<union of 104 residue classes (mod 32400)>
<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)>
<union of 189 residue classes (mod 129600)>
<union of 160 residue classes (mod 129600)>
<union of 136 have to treat 4 separately
<union of 133 residue classes (mod 129600)>
<union of 122 residue classes (mod 129600)>
<union gap>1^(u**^2a-*u)
<union of 106 residue classes (mod 129600)>
<union of 104 residue classes (mod 129600)>
<union of 100 residue classes (mod 129600)>
<union of 96 residue classes (mod 129600)>
<union of 60 residue classes (mod 129600)>
<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)>
<union of 32 residue classes (mod 129600)>
<union of 24 residue Now we know that any positive integer greater than 1 lies in the same
<union of 16 residue classes (mod 129600) (10 classes)>
<nionof 12 (od129600)java.lang.StringIndexOutOfBoundsException: Range [42, 43) out of bounds for length 42
<union of 10 java.lang.StringIndexOutOfBoundsException: Range [68, 17) out of bounds for length 68
<union of 8 residue classes (mod 129600)>
<union of 6 residue classes a ,we essentiallyrepeat the 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 classes (mod 777600) (6 classes)>
<union of 30 residue classes (mod 3110400)>
<union of 26 residue classes (mod 3110400)>
<of classes (mod 3110400)>
<> l->PositionSublist(l=);
<union of 14 residue classes (mod 3110400) (8 java.lang.StringIndexOutOfBoundsException: Index 51 out of bounds for length 17 705406(777600) U 2007076(3110400) U 2649406(3110400) U 2784676(3110400)
<union of 14 residue classes (mod 9331200) (8 classes)> 1483006(2332800) U 2649406(9331200) U 2784676(9331200) U 5117476(9331200)
<union of 16 residue classes (mod 27993600) (6 classes)> 2784676(9331200) U 5117476(9331200)
[ java.lang.StringIndexOutOfBoundsException: Index 4 out of bounds for length 4
gap> Maximum(List(stabelm{[1..maxind]},
> f->Maximum(gap> (); 37387
ap = 537387;java.lang.StringIndexOutOfBoundsException: Index 29 out of bounds for length 29
gap> for i in [1g>SortParallel(decs,stabelm,(,)
> 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 java.lang.StringIndexOutOfBoundsException: Range [34, 36) out of bounds for length 8
<Example>
<![CDATA[
gap> phi := EpimorphismFromFreeGroup(G);
[ a, u ] -> [ a, u ]
gap> F := Source(phi);
<free group unionof30 residueclasses (od 720>
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^-union 28 classes mod5760)java.lang.StringIndexOutOfBoundsException: Index 40 out of bounds for length 40
a^-1*u^2*a, a^-1*u^-6*a, a^union java.lang.StringIndexOutOfBoundsException: Range [21, 20) out of bounds for length 40
a^-1*u^-2*a^-3*u^4*a^2, a^2*(a*u^2)^2, (a*u^-4)^2*a^-2,
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 Label="sec:ThreeButNotFourTransitiveGroup">
<Heading>
A group which acts 3-transitively, but not 4-java.lang.StringIndexOutOfBoundsException: Index 53 out of bounds for length 42
</
In this section, weunion of16residue java.lang.StringIndexOutOfBoundsException: Range [29, 28) out of bounds for length 55
by the two permutations <M>n \mapsto n + 1</M> union of 12 residue classes (mod 259200) (10 classes)>
acts 3-transitively, but not 4-transitively on the set of integers.
<Example>
<![CDATA[
gap> G := Group(ClassShift(0,1),ClassTransposition(1,2,0,4));
<rcwa group over Z with 2 generatorsfor in[.Lengthstabelm]do
IsTame(G;
false
>(1-*.22)3(.^*.)3 < free product C_infty*java.lang.StringIndexOutOfBoundsException: Index 73 out of bounds for length 73
IdentityMapping( Integers )
gap> Display(G:CycleNotation:=false);
/
| 2n-2 if n in
n |-> < (n+2)/2 if n in 0> l: 123;
|n n in 24)
\
]
]]>
</Example>
This group acts transitively on &ZZ;, since already the cyclic
group generated by the first> for in stab[]do
show that it acts 2-transitively. We essentially proceed as in the example
in the previous section, by checking that the stabilizer ofjava.lang.StringIndexOutOfBoundsException: Index 59 out of bounds for length 30
acts transitively on <M>&ZZ; \setminus \{0\}</M>.
<Example>
<![CDATA[
Transposition(,04,
> ClassShift(0,1 odjava.lang.StringIndexOutOfBoundsException: Index 8 out of bounds for length 8
gap> tups := Concatenation(List([1..6],k->Tuples([-1,0,java.lang.StringIndexOutOfBoundsException: Range [3, 4) out of bounds for length 3
gap> tups := Filtered(tups,tup->ForAll([[0,0],[-1,1],[1,-1]],
> l->PositionSublist(java.lang.StringIndexOutOfBoundsException: Index 60 out of bounds for length 22
gap> Length(tups); 189
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> stabelm := List(stab,tup->Product(List(tup,i->gens[i+2])));;
gap> Z \ 1(8) U 7 () 78
[[4 6] [8,4][ 163]]
gap>decs: List(,);
(4,3() (4, (4,2(),04),4(,2(4),4,0) () 0(8), 3(8) ]
gap> Union(decs);
Integers
]]>
</Example>
as in previoussection,itremainsto checkthat the
<>/>value all lie in orbit containing
ofn;0java.lang.StringIndexOutOfBoundsException: Index 39 out of bounds for length 39
gap> S :unionof 116 residueclasses(mod32400)
>for elminstabelmdo S =Union(Selm,^(elm-); odjava.lang.StringIndexOutOfBoundsException: Index 62 out of bounds for length 62
gap> IsSubset(S,Difference([-21..21],[0])); # Not yet ..
false
gap> for elm in stabelm do S := Union(S,S^elm,S^(elm^-1)); od;
gap> IsSubset(S,Difference([-21..21],[0])); # ... but u 189residueclasses)>
true
]]>
</Example>
Now we have to check for 3-transitivity. Since we cannot find for every
residue class an element of the pointwise stabilizer of <M>\{0,1\}</M>
whichunion of104 residue (129600)java.lang.StringIndexOutOfBoundsException: Index 43 out of bounds for length 43
subtractions into consideration. union of 96 residue classes (mod 129600
elementsare uite small,simply sets representatives :
<Example>
<![CDATA[
gap> tups := Concatenation(List([1..10],k->Tuples([-1,0,1],k)));;
gap>unionof 32residueclasses ( )
> l->PositionSublist(tup,l)=fail));;
gap> Length(tups); 3069
gap> 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 = [0,1] then Add(stab,tup); fi;
> od;
gap>Length(tab)java.lang.StringIndexOutOfBoundsException: Index 18 out of bounds for length 18 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
decsp =(tabelm,lm-Filtered[.16,n-nelmn);
[ [ 9,13 ] [10, ,14 ,[1216,[ , , ,16 ,
[ 9, 11, 13, 15 ], [ 9, 11, 13, 15 ], [ 12, 16 ], [ 12, 16 ],
, ,13,15 ]
gap Union(ecsp);
[ 9 .. 16 ]
gap> decsm := List(stabelm,elm->Filtered([-16..-9],n->n^elm>n));
[ [ -15, -13, -11, -9 ], [gap> for iin[.maxind do
[ -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];;
gap> for elm in stabelm do S := Union(S,S^elm,S^(elm^-1)); od;
gap> IsSubsetgap>smallnum;
true
]]>
</Example>
At this point we have established 3-transitivity.
Itremains to the M><M does not act 4-.
We do
Since <M>n</M> mod 8 determines the image of <M>n</M> under
a generator of <M>G</M> (mod 4), it suffices to compute (mod 8):
<Example>
<!<ree group onthe
gap> orb := [[0,1,2,3]];;
gap> extend := function ()
> local gen;
> for gen in gens do
> orb := Union(orb,List(orb,l->List(l,n->n^gen) mod 8));
> java.lang.StringIndexOutOfBoundsException: Index 10 out of bounds for length 10
> ;
gap> repeat
old =ShallowCopy)java.lang.StringIndexOutOfBoundsException: Index 31 out of bounds for length 31
> extend(); Print(Length(orb),"\n");
> until orb = old; 7 27 97 279 573 916 1185 1313 1341 1344 1344
gap> Length(Set(List(orb,l->l mod 4))); 120
gap> last < 4^4;
true
]]>
</Example>
hows that<><M> 4transitively on&;&ZZ.
The corresponding calculation for 3-tuples looks as follows:
Needless to say that the latter kind of argumentation is not suitable
for proving, but only for java.lang.StringIndexOutOfBoundsException: Index 34 out of bounds for length 3
<Section Label="sec:SlowlyContractingMapping">
<Heading>
An rcwa mapping which seems to be contracting, but very slow
</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 java.lang.StringIndexOutOfBoundsException: Index 32 out of bounds for length 9
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 proved so far!):
<Example>
<[[
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
See ?LikelyContractionCentre for more information.
[ -136, -91, -82, -68, -61, -55, -41, -37, -34, -25, -17, -10, -7, -5,
-1, 0, 1, 2 ]
gap> S0^T = S0; # This holds by definition of the contraction centre.
true
gap> List([1..30],n->Length(Trajectory(T,n,S0)));
[ 1, 1, 5, 2, 4gap>stabelm: List(,-ProductListtup-gens[+]);; 11, 11, 8, 16, 8, 70, 13, 13, 13 ]
gap> Maximum(List([1..1000],n->Length(Trajectory(T,n,S0)))); 113
ap>Maximum(List(.1000]n-Maximum((T,,S0))); 125252
]]>
</Example>
The following mapping seems to be contracting as well, but its trajectories
are much longer:
/
| n/6 if n in 0(6)
| (5n+1)/6 if n in 1(6)
| (7n-2)/6 if n in 2(6)
|><(1n+)6ifnin 36java.lang.StringIndexOutOfBoundsException: Index 32 out of bounds for length 32
| (11n-2)/6 if n in 4(6)
| (11n-1)/6 if n in 5(6)
|
\
gap> S0 := LikelyContractionCentre(f6,1000,100000);;
#I Warning: `LikelyContractionCentre' is highly probabilistic.
The returned result can only be regarded as a rough guess.
See ?LikelyContractionCentre for more information.
gap> Trajectory(f6,25,S0);
[ 25, 21, 39, 72, 12, 2 ]
gap> List([1..
[ 1, 1, 3, 4, 1, 2, 3, 2, 1, 5, 7, 2, 8, 17, 3, 16, 1, 4, 17, 6, 5, 2, 5, 5, 6, 1, 4gap>tupstupstup-ForAll[[,][11,1,1]java.lang.StringIndexOutOfBoundsException: Index 61 out of bounds for length 61 19, 23517, 3, 9, 3, 1, 18, 14, 2, 20, 23512, 14, 2, 6, 6, 1, 4, 19, 12, 23511, 8, 23513, 10, 1, 13, 13, 3, 1, 23517, 7gap> : [; 12, 8, 6, 18, 14, 23516, 31, 12, 23545, 4, 21, 19, 5, 1, 17, 17, 13,
java.lang.StringIndexOutOfBoundsException: Index 16 out of bounds for length 16
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 java.lang.StringIndexOutOfBoundsException: Range [0, 73) out of bounds for length 1
point 2 after 19949562 steps. <P/>[[ ,13 ,[10 ,14, 16,[,16] ,13] [1216],
When constructing the mapping <C>f6</C>, the denominators of the
partial mappings have been chosen to be equal and the numerators have
been chosen to be numbers coprime to the common denominator, whose product
wer 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>f6</C>, it seems likely that this does not make the problem of deciding
whether the mapping <C>T</C> is contracting [[-,-,-1, 9] [-, 12],[-, - ] [ -,-1]
even for mappings with much shorter trajectories than <C>T</C>
the problem seems to be equally hard. A solution can usually only be found
in trivial cases, i.e. for example when there is some <M>k</M> such that
applying the <M>k</M>th power of the respective mapping to any integer
decreases its absolute value. <P/>
Enter<>AssignGlobals(oadRCWAExamples(.);<C
in order to assign the global variables defined in this section.
<ectionLabel=sec:ndaloroResult>
<Heading>Checking a result bya generatorof <></>(od ) sufficesto compute m&;8)java.lang.StringIndexOutOfBoundsException: Index 74 out of bounds for length 74
In <Cite Key="Andaloro00"/>, P. Andaloro has shown that java.lang.StringIndexOutOfBoundsException: Index 65 out of bounds for length 25
trajectories of integers <M>n \in 1(16)</M> under the Collatz mapping always
contain 1 would be sufficient to prove the <M>3n+1</M> conjecture.
nthis java.lang.StringIndexOutOfBoundsException: Range [27, 26) out of bounds for length 67
union of the <script type="text/>java.lang.StringIndexOutOfBoundsException: Range [46, 45) out of bounds for length 57
mapping <M>T</M> contains <M>& od;
Thusweput<> = (<enough. Thus we put <M>S := 1(16)</M
set&java.lang.StringIndexOutOfBoundsException: Range [1, 8) out of bounds for length 6
<
!CDATA
gap>
< 2
gap> S (16
gap> :UnionSjava.lang.StringIndexOutOfBoundsException: Index 4 out of bounds for length 4 1(16) U2(24)
:Union(SSjava.lang.StringIndexOutOfBoundsException: Index 4 out of bounds for length 4
s "APprompt>&t;/span> <java.lang.StringIndexOutOfBoundsException: Range [45, 44) out of bounds for length 95
ionS^);
<union of 30 residue classes (mod 144)>
>S:UnionS^,
< of classes (od )java.lang.StringIndexOutOfBoundsException: Index 39 out of bounds for length 39
gap>S = SS;
<union of 172
gap> S := Union(S,S^T);
<union of 676 residue java.lang.StringIndexOutOfBoundsException: Index 26 out of bounds for length 0
UnionSS)
< of residueclasses( )>
gap> span=>&;/
<of 2638residue classes(od 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
]]>
java.lang.StringIndexOutOfBoundsException: Index 11 out of bounds for length 10
Furtherjava.lang.StringIndexOutOfBoundsException: Index 2 out of bounds for length 2 Section <Ref Label="sec:java.lang.StringIndexOutOfBoundsException: Index 49 out of bounds for length 3
Enter <C>AssignGlobals(LoadRCWAExamples().CollatzMapping);</C>
in order to assign the global variables defined in this section.
<SectionLabel="sec:MatthewsLeighExamples">
<Heading>Two examples by Matthews and Leigh</Heading>
In <Cite Key="MatthewsLeigh87"/>, K. R. Matthews and
G. M. Leigh have shown that two trajectories of the following
(surjective, but not injective) mappings are acyclic (mod <M>x<java.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0
and divergent:
<Example>
<![CDATA[
java.lang.StringIndexOutOfBoundsException: Range [32, 3) out of bounds for length 50
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);
mapping GF()x]with modulusx
[ -,-1, -82 -686155-,-, 34 -25 -17, -, 7,-
| P/x if P in 0(x) P | -,0,1,2]
|
\
The pattern which Matthews and Leigh used to show the divergence of the
above trajectories
Markovchainswith the two 0nbsp;mod&;<x/>and 1 mod&# Warning `'highly probabilistic.
This looks clearly acyclic, thus the trajectories diverge.
Needless to say<union of residueclasses mod144>
the proof along these lines given in the article cited above, but just
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 Label="sec:OrdersOfCommutators">
<Heading>Orders of commutators</Heading>
We enter some wild rcwa permutation:
<Example>
<![CDATA[
gap> u := RcwaMapping([[3,0,5],[9,1,5],[3,-1,5],[9,-2,5],[9,4,5]]);;
gap> IsTame(u);;
gap> Display(u);
Wild rcwa permutation of Z with modulus 5
/
| 3n/5 if n in 0(5)
| (9n+1)/5 if n in 1(5)
n |-> < (3nin order to assign the globalvariables ithis section
| (9n-2)/5 if n in 3(5)
| (9n+4)/5 if n in 4(5)
]]>
</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<G. M. eigh shownthat trajectories the
<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>ap>traj1modx =TrajectoryML1,ne(),,);java.lang.StringIndexOutOfBoundsException: Range [48, 49) out of bounds for length 48
<![CDATA[
gap> x := Indeterminate(GF(2));; SetName(x,"x");
gap> R := PolynomialRing(GF(2),1);
GF(2)[]
gap> a := ClassTransposition(0,x,1,x);;
gap>b :=ClassTransposition(,x2,1x2);
c :=1,,0x2+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
java.lang.StringIndexOutOfBoundsException: Index 8 out of bounds for length 1
Rcwa permutation of GF(2)[x]gap> List[(RR,neR)]val-)java.lang.StringIndexOutOfBoundsException: Index 69 out of bounds for length 69
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 1, 0,1, , ,0,11 , ,1 ,1 ,111,1 ,1 1,11
|
\
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)
\
]
]]>
</Example>
We can easily find 2 normal subgroups of <C>G</C>:
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, ]]java.lang.StringIndexOutOfBoundsException: Range [3, 4) out of bounds for length 3
local words, gens, tuples, w;
=()java.lang.StringIndexOutOfBoundsException: Range [33, 34) out of bounds for length 33
tuples := EnumeratorOfTuples([1..3],lng);
words := [];
for w in tuples do
if (w[1] = 1 or not 1 in w)
and PositionSublist(w,[1,1]) = 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,java.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0
then Add(words,w); fi;
od;
words := List(words,word->Product(List(word,i->gens[i])));
return words;
end;
]]>
</Listing>
Now let's compute the possible orders of products of 3, 5, 7 or 9 generators:
<Log>
<![CDATA[Wild rcwa permutationof modulus5
gap> F := FreeGroup("a","b","c");;
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) ]
gap>B3 : F3;
[a**cc java.lang.StringIndexOutOfBoundsException: Index 9 out of bounds for length 9
gap> B3 := List(B3,g->g^phi);
[ <rcwa permutation of GF(2)[x] java.lang.StringIndexOutOfBoundsException: Index 35 out of bounds for length 10
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 gap> nu := ClassShift(0,1);; # n -> n + 1
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)[x] with modulus ,626365,66, 69 , 73 , , 80, ,848587
<88 , ,93
<rcwa gap> List(l,k->Order(Comm(u,nu^k)));
<rcwa permutation of GF(2)[x] with modulus x 13infinity7 , ,5 , infinity,
gap> List(B7,Order);
[ 12, 12, 12, 30 ]
gap> B9 := NormedWords(F,9);
[ ****bcabc a**c****a*c ***a****bc ***aac***ac,java.lang.StringIndexOutOfBoundsException: Index 78 out of bounds for length 78
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().OddNumberOfGens_FiniteOrder);</C>
in order to assign the global variables defined in this section.
<Section Label="sec:AbelianGroupOverPolynomialRing">
<Heading>An abelian rcwa group over a polynomial ring</Heading>
We enter a 2-generated abelian wild rcwa group over GF(4)[<M>x</M>]:
Example>
<![CDATA[
gap> x := Indeterminate(GF(4),1);; SetName(x,"x");
gap> R := PolynomialRing(GF(4),1);
GF(2^2)[x]
gap> e := One(GF(4));;
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 GF(2^2)[x] with modulus x^2+x+Z(2^2)^2>
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 that all orbits on GF(4)[<M>x</M>] under the action
of <C>G</C> are finite. <P/>
Now we compute the action of the group <C>G</C> on one of its orbits, and
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,
4
^x^+^++ ^+x^2+(^2)* x^5+^Z(^2^2*Z22^2]
gap> H := Action(G,orb);
Group([ (1,2,4,7,6,9,12,14)(3,5,8,11,10,13,15,16),
(1,3,6,10)(2,5,9,13)(4,8,12,15)(7,11,14,16) ])
gap> IsAbelian(H); # check ...
true
gap> IsCyclic(H); # H, and therefore also G, is not cyclic
false
gap> Exponent(H); 8
gap> Collected(List(ShortOrbits(G,AllResidues(R,x^4),100),Length));
[ [ 1, 4 ], [ 2, 6 ], [ 4, 12 ], [ 8, 24 ] ]
java.lang.StringIndexOutOfBoundsException: Index 3 out of bounds for length 3
</Example
Changing the generators a little changes the structure of the group
and its action on the underlying ring a lot:
<SectionLabel="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</Agap> a =ClassTransposition(0,x,1x); a non-solvable finite permutation group:
<Example>
<![CDATA[
gap> a := RcwaMapping([[3,0,2],[3, 1,4],[3,0,2],[3,-1,4]]);;
gap> b := RcwaMappingjava.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0
gap> G := Group(a,b);;
gap> ShortOrbits(Group()[] P- P Z2^0
[ [ -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
[ 7, 5, 6, 4, 10, 15 ]
gap> U := Group(Comm(a,b),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>
degree 6. Therefore the group <A>G</A> is not solvable.
We conclude this example by factoring the group element <A>elm</A> into
generators:
<Example>
<![CDATA[
gap> F := FreeGroup("a","b");
<group on the generators [a ]
RepresentativeActionPreImageG,S,(,1,),nTuples,; 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 = elmgap> /;
true
]]>
</Example>
Enter <C>AssignGlobals(LoadRCWAExamples().CheckingForSolvability);</C>
in order to assign the global variables defined in this section.
<SectionLabel="sec:LocalExample">
<Heading>Some examples over (semi)localizations of the integers</Heading>
start with something canobserve when trying to Qtransfer<Q>an
rcwa mapping from the ring of integers to one of its localizations:
<Example>
<![CDATA[
gap> a := RcwaMapping([[3,0,2],[3,1,4],[3,0,2],[3,-1,4]]);;
gap> IsBijective(a);
true
gap> a2 := LocalizedRcwaMapping(a,2);
<rcwa mapping of Z_( 2 ) with modulus 4>
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&<>&;{(2}/.
Moving to <M>&ZZ;_{(2,3)}</M> java.lang.StringIndexOutOfBoundsException: Index 33 out of bounds for length 0
<Example>
<![CDATA[
gap> a23 := SemilocalizedRcwaMapping(a,[2,3]);
<rcwa mapping of Z_ and PositionSublist(,11)=
gap> IsBijective(a23);
true
]]>
</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:
<SectionLabel="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.
Rcwa permutation of Z with modulus 32, of order 257
| 16n+18 if n in 1(2) \ 31(32)
| n+16 if n in 2(32) U4(32) U6(32) U8(32) U10(32) U
| 12(32) U14(32)
| n-14 if n in 18(32) U20(32) U22(32) U24(32) U26(32) U
n |-> < 28(32) U30(32)
| 16n+2 if n in 0(32)
| n/16 if n in 16(32) 313132java.lang.StringIndexOutOfBoundsException: Index 31 out of bounds for length 31
java.lang.StringIndexOutOfBoundsException: Range [9, 10) out of bounds for length 9
\
gap> Display(x_257);
Rcwa permutation of Z with modulus 32, of order 257
<SectionLabel="sec:[ 1, 16, 1], [ 16, 18, 1[ ],[16, ,1,
<Heading> Images and preimages under the Collatz[,16, ] ,181, 1, 16, 1] ,18 ]java.lang.StringIndexOutOfBoundsException: Range [67, 68) out of bounds for length 67
We have a look at the images of the residue class 1(2)
under powers of the Collatz mapping.
Thus the image gets stable after applying the 032,2) 512),4(), 20(512),(512) 22(),
the 6thtime.HenceMT^<M>mapstheresidue 1()surjectively
onto the union of the residue classes 1(3) and 2(3), which <M>T</M 30() (512),1() 34512, 50(),36() 52()
stabilizes setwise.
Now we would like to determine the 60(512), 46(512), 62(512), 48(512)3(32), 66(512) 82(),java.lang.StringIndexOutOfBoundsException: Index 63 out of bounds for length 63
under <M>T^6</M>. The residue class 1(2) has to be the disjoint union of
these sets.
The images of the residue class 0(3) under powers of <M>T</M> look
as follows:
<Example>
<![CDATA[
gap> S0 := ResidueClass(Integers,3,0); 03)
gap> S1 := S0^T; 0(3) U5(9)
ap S2 :=S1^;
()U () U (98(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) U37(81)
gap> S6 := S5^T;
Integers
gap> S7 := S6^java.lang.StringIndexOutOfBoundsException: Range [0, 15) out of bounds for length 0
Integers
]]>
</Example>
Thus every integer is the image <!-- #############################################--
under
<M>3n+1</M> conjecture for multiples of 3.
We can obtain the corresponding result for multiples of 5 as follows:
<Example>
<![CDATA[
gap> S := [ResidueClass(Integers,5,0)];
[ 0(5) ]
gap> for i in [1..12] do Add(S,S[i]gap> =RcwaMapping([,02],[3, 1,4],[3,,]3-14]);
gap> for s in S do View(s); Print("\n"); od; 0(5) 0(5) U8(15) 05(15U815java.lang.StringIndexOutOfBoundsException: Index 20 out of bounds for length 20 0(5) U2(15) U4(15) U8(15) U29(45)
<union of 73 residue classes infinity,infinity ]
<union of 244 residue classes (mod 405)>
<nion of 784residue classes ( 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) U2(3) U0(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.
Often, dealing with rcwa permutations is easier.
Indeed the Collatz mapping <M>T</M> can be extended in natural
ways to permutations of <M>&ZZ;^2</M>. For example, the following
permutation acts on the second coordinate just like <M>T</M>:
Rcwa permutation of Z^2 with modulus (1,0)Z+(0,6)Z
/
| (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 gap> =S0T;
| (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 (m,n) in (1,2)+(2,0)Z+(0,3)Z
|
\
]]>
</Example>
Now, the <M>3n+1</M> conjecture is equivalent to the assertion that
the line <M>n=4</M> is a set of representatives for the cycles of
<C>Sigma&uscore;T</C> on the half plane <M>n > 0</M>. <P/>
Let's have a look at a part of a cycle of <C>Sigma&uscore;T</C>:
Whileit makeconjectures regardingthebehaviourofcycles
<Sigma&;<C> obtainingresults it is apparently hard.
We observe however thatunionof 21 residue classesmod 64>
of two permutations of <M>&ZZ;^2</M> whose cycles can be described
easily:
<Example>
<![CDATA[
gap> a := RcwaMapping(Integers^2,[[1,0],[0,2]],[[[[4,0],[0,1]],[0, 0],2],
> [[[4,0],[0,1]]gap> AsUnionOfFewClasses(U);
<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);
cwa of (10)Z+0,2Z
/
| (2m,n/2) if (m,n) in (0,0)+(1,0)Z+(0,2)Z
(m,n) |-> < (2m+1,(n-1)/2) if (m,n) in (0,1)+(1,0)Z+(0,2)Z
|
\
gap> Display(b);
Rcwa permutation of Z^2 with modulus (2,0)Z+(0,3)Z
/
| (m,3n+2) if (m,n) in (1,0)+(2,0)Z+(0,1)Z
| (m/2,n) if (m,n) in (0,0)+(2,0)Z+(0,3)Z U
(m,n) |-> < (0,1)+(2,0)Z+(0,3)Z
| (m,n) if (m,n) in (0,2)+(2,0)Z+(0,3)Z
|
\
]]>
</Example>
It is03)U5)
The cycles of <C>a</C> have roughly hyperbolic shape and run, so to
speak, from <M>(0,\pm \infty)</M> to <M>(\pm \infty,0)</M>. A given cycle contains only finitely many points both of whose coordinates
are nonzero. The fixed points of <C>a</C> are (0,0) and (-1,-1).
We have a look at an example of a cycle of <C>a</C>:
<SectionLabel="sec:GrigorchukGroups">
<Heading>
Finite quotients of <!-- #################################################################### -->
</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 &RCWA;.
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:
<
<![java.lang.StringIndexOutOfBoundsException: Index 5 out of bounds for length 0
SequenceElement := function ( r, level )
The actual constructors for the generators are modified as follows:
<Listing>
<[CDATA[
:= -> Permutation(ClassTransposition(0,2,1,2),[0..2^level-1]); b := level -> SequenceElement(0,level);
c := level -> SequenceElement(Example>
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"/>:
<>
<![CDATA[
gap> FourCycle := RcwaMapping((4,5,6,7),[4..7]);
( 04,14,2() 3()java.lang.StringIndexOutOfBoundsException: Index 26 out of bounds for length 26
gap> GrigorchukGroup2Generator := function ( level ) if level then returnFourCycle
> return
> * 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:
<Example>
<![CDATA[
gap> G := List([1..4],lev->GrigorchukGroup2(lev)); # The first 4 quotients.
[ <rcwa group over Z with 2 generators>,
<rcwa group over Z with 2 generators>,
rcwa group over Z 2generators>
<rcwa group over Z with 2 generators> ]|
gap> H := List([1..4],lev->Action(G[lev],[0..4^lev-1])); # Isom. perm.-gps.
[ Group([ (1,2,3,4), (1,2,3,4) ]),
Group([ (1,2,3,N, the M3n1/> tothe assertion
(1,,,13)2,,,)(4,8,,16)])
<permutation group with 2 generators>,
<permutation group with 2 generators> ]
gap>List(,Size);
[ 4, 1024, 4294967296, 1329227995784915872903807060280344576 ]
gap> List(last,n->Collected(Factors(n)));
[ [ [ 2, 2 ] ], [ [ 2, 10 ] ], [ gap>Trajectory([0,7]75)
(NilpotencyClassOfGroup)
[ 1, 6, 14, 40 ]
]]>
</Example>
Enter [ 263] 32767395 ],[65535 ,[131071,890 ,
order toassignthe definedthissection
<Section Label="sec:ForwardOrbit">
<Heading>
Forward orbits of a monoid with 2 generators
</Heading>
The <M>3n+1</M> conjecture asserts that the forward orbit of any positive
under Collatz mapping MT/M>containsnbsp1 contrast,
ories thetwo mappings
<Alt Only="LaTeX">
<Display>
<![CDATA[T_5^\pm: \ \mathbb{Z} \longrightarrow \mathbb{Z}, \ \ \ , 8 ] 281474439704249 ,[ , ,
n \ \longmapsto \
\begin{cases}
\frac{n}{2} & \text{if} \ n \ \text{even}, \\
\frac{5n \pm 1}{2} & \text{if} \ n \ \text{odd}
\end{cases}]]>
</Display>
</Alt>
<Alt Only="HTML"><![CDATA[<center>
width=""61java.lang.StringIndexOutOfBoundsException: Index 51 out of bounds for length 51
alt = "T5+/-: Z -> Z, n |-> (n/2 if n even, (5n+/-1)/2 if n <Example>
</center>]]></Alt>
<Alt Only="Text"><Verb><![CDATA[
/
| n/2if n even,
T_5+/-: Z -> Z, n |-> <
| (5n +/- 1)/2if n odd
\
]]></Verb></Alt>
diverge.
However we can showgap> a := RcwaMappin(2,[,][,2],[[,][,]],0,0]2,
positive integer under the action of the monoid generated by the two
mappings <M>T_5^-</M> and <M>T_5^+</M> indeed contains 1.
First of all, we enter the generators:
We look for a number <M>k</M> such that for any residue class <M>r(2^k)</M>
there is a product <M>f</M> of <M>k</M> mappings <M>T_5^\pm</M>
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(decr);
> Print(k,": "); View(decreasable); Print("\n");
> k := k + 1;
> until decreasable = Integers; 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 \73)U (32) 6: <union of 48 residue classes (mod 64)|(m2n if (n)in 00+20Z(,) U 7: Integers
]]>
</xample
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 java.lang.StringIndexOutOfBoundsException: Index 28 out of bounds for length 0
ditionfor<>mall<Q>nbsp;Mn/> ncase<>><M have
M(+b)c\geq n</>ifonlyif<M>n \eqb(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 are nonzero. The fixed points of <C>a</C>,0and -1-1.
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&1by successively computing preimages of&;:
<Example>
<![CDATA[
gap> S := [1];; k := 0;;
gap> repeat
> S := UnionI is as an exercise to reader find out how the
> : k1java.lang.StringIndexOutOfBoundsException: Index 16 out of bounds for length 16
untilIsSubsetS,)java.lang.StringIndexOutOfBoundsException: Index 29 out of bounds for length 29
gap> k; 17
]java.lang.StringIndexOutOfBoundsException: Index 3 out of bounds for length 3
</Example>
Enter <C>AssignGlobals(LoadRCWAExamples().CollatzMapping);</C>
in order to assign the global variables defined injava.lang.StringIndexOutOfBoundsException: Index 50 out of bounds for length 0
The free group of rank 2 embeds into RCWA(&ZZinvestigated in givenon the&;website -see
inthesubgroup whichi generatedbyallclass .
An explicit embedding can be constructed by transferring the construction
of the so-called <Q>Schottky groups</Q> (cf. <Cite Key="LaHarpe00"/>,
page 27) from PSL(2,&CC;) to RCWA(&ZZ;)
(we use the notation from the cited book):
<Example>
<![CDATA[
gap> D := AllResidueClassesModulo(4);
[ 0(4), 1(4)
gap gamma1: (Integers)java.lang.StringIndexOutOfBoundsException: Index 51 out of bounds for length 51
> Difference(Integers,D[1]),D[2]);;
gap> gamma2 := RepresentativeAction(RCWA(Integers),
> Difference(Integers,D[3]),D[4]);;
gap> F2 := Group(gamma1,gamma2);
<rcwa end;
]]>
</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 3class transpositions, java.lang.StringIndexOutOfBoundsException: Range [0, 59) out of bounds for length 10
Attr""
Label="for a group"/> to embed free groups of any rank <M>\geq 2</M>.
P>
We give another only slightly different representation of the free group
of rank 2. We verify that it really is one by applying the
so-called <E>Table-Tennis Lemma</E> (see e.g. <Cite Key="LaHarpe00"/>,
Section II.B.) to the infinite cyclic groups generated by the two
generators and to the same two sets <C>X1</C> and <C>X2</C> as above:
Drawing the transition graphs of <C>r1</C> and <C>r2</C> for modulus 4
may helpto whatis actually done .
It is easy to see that\rac{n\pm1}2 \text{f}\n\\text{odd}
is <E>not</E> free:
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:
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 Welookfor number<>/>such that foranyresidueclass<>(2k</>
());
<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 ">nbsp;/java.lang.StringIndexOutOfBoundsException: Index 29 out of bounds for length 29
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