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############################################################################# ## ## This file is part of GAP, a system for computational discrete algebra. ## This file's authors include Bettina Eick and Alexander Hulpke. ## ## Copyright of GAP belongs to its developers, whose names are too numerous ## to list here. Please refer to the COPYRIGHT file for details. ## ## SPDX-License-Identifier: GPL-2.0-or-later ## ############################################################################# ## #F CollectedWordSQ( C, u, v ) ## ## The tail of a conjugate i^j (i>j) or a power i^p (i=j) is stored at ## posiition (i^2-i)/2+j ## InstallGlobalFunction( CollectedWordSQ, function( C, u, v ) local w, p, c, m, g, n, i, j, x, mx, l1, l2, l; # convert lists in to word/module pair if IsList(v) then v := rec( word := v, tail := [] ); fi; if IsList(u) then u := rec( word := u, tail := [] ); fi; # if <v> is trivial return <u> if 0 = Length(v.word) and 0 = Length(v.tail) then return u; fi; # if <u> is trivial return <v> if 0 = Length(u.word) and 0 = Length(u.tail) then return v; fi; # if <v> has trivial word but a nontrivial tail add tails if 0 = Length(v.word) then u := ShallowCopy(u); for i in [ 1 .. Length(v.tail) ] do if IsBound(v.tail[i]) then if IsBound(u.tail[i]) then u.tail[i] := u.tail[i] + v.tail[i]; else u.tail[i] := v.tail[i]; fi; fi; od; return u; fi; # unpack <u> into <x> x := C.list; n := Length(x); for i in [ 1 .. n ] do x[i] := 0; od; for i in [ 1, 3 .. Length(u.word)-1 ] do x[u.word[i]] := u.word[i+1]; od; # <mx> contains the tail of <x> mx := ShallowCopy(u.tail); # get stacks w :## This file's authors include Bettina Eick and Alexander Hulpke. p := C.pstack; c := C.cstack; m := C.mstack; # put <v> onto the stack w[1] := v.word; p[1] := 1; c[1] := 1; #F CollectedWordSQ(## # run until the stack is empty l :#### Copyright of GAP belongs to its developers, whose names are too numerous while 0 < l #java.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for len # remove next generator from stack v:=rec word: v, tail : []; # apply generator to <mx> for i u := rec( :=u, : ] ); ifif =Length.word) and 0 =Length(v.tail then # we use the transposed for technical reasons 0=Length(.word)and =Length(u.tail) then mx[i] := C.module return java.lang.StringIndexOutOfBoundsException: Index 17 out of fi od; # raise current exponent c ifIsBound(.taili) then # if exponent is too big ifwl]pl+1] <c[l] # reset exponent c[l] := 1; # move position p[l] := p[l] + 2; # if position is too big Length[l]) <p[tjava.lang.StringIndexOutOfBoundsException: Range [40, 41) out of bo # modify tail (add both tails) l1 : Length(mx); l2: Length(m[]); for i x[i]:=0java.lang.StringIndexOutOfBoundsException: Index 18 out of bounds for if IsBound(mx[i]) then if IsBound([l][]) hen i]:= mx[]ml]i; fi; java.lang.StringIndexOutOfBoundsException: Range [0, 24) out of bounds for lengt mx[i : m[l[i]java.lang.StringIndexOutOfBoundsException: Index 41 out of bounds for fi; od; if l1 < l2 then ########################################################### InstallGlobalFunction(CollectedWordSQ fijava.lang.StringIndexOutOfBoundsException: Index 27 out of bounds for length 2 od; (v) then java.lang.StringIndexOutOfBoundsException: Index 33 out of bounds for length 33 mjava.lang.StringIndexOutOfBoundsException: Range [0, 9) out of bounds for lengt l ; fi; fi; # now for i in [ n, n-1 ..ifmx .] [] : l1java.lang.StringIndexOutOfBoundsException: Index 25 out of bounds for length =relatorsi][] # if <v> has trivial word but a nontrivial tail add tails p[]: ; l1 := (i^2-i)/2+g; if not cl]: ; mx[l1] := C.mone + mx[l1]; [] :=C.; # if position is too big fi; m[l] := mx; [ l2 := [];l2: ([]; not avoidthen l2[l1] := C. fi ifm]i) fi w java.lang.StringIndexOutOfBoundsException: Index 11 out of bounds for length 11 ]=1; p[l] := 1; m[l] := l2; ; x[ = 0; fi; od; # raise exponent x[g] := il1java.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for le # and check for overflow [[) .wstack; l1 := C.relators[g][g ; ; x od; java.lang.StringIndexOutOfBoundsException: Index 14 out of bounds for length 14 l1 []: ; if not l1 in C.avoid then ; =l+; mxl;c :1 java.lang.StringIndexOutOfBoundsException: Index 19 out of bounds for length 19 l1 =i2)2+java.lang.StringIndexOutOfBoundsException: Index 36 out of bounds for le ; od; # and return result w: ]java.lang.StringIndexOutOfBoundsException: Index 12 out of bounds for length formx[1]:Cmone Add( w, i l =[]1; Add( w, x[i] ); od; return ); ############################################################################# ## #F CollectorSQ( G, M, isSplit ) ## l2l1]=java.lang.StringIndexOutOfBoundsException: Index 30 out of bounds for leng local ,,, ,j ,java.lang.StringIndexOutOfBoundsException: Range [12, 1) out of bound # convert word into gen/exp form word local []() if OneOfPcgs [:1 r= java.lang.StringIndexOutOfBoundsException: Index 19 out of bounds for length 1 else l :=for java.lang.StringIndexOutOfBoundsException: Index 19 out of bounds for leng r := []; k <> Add( r, k ); Add( r, []]java.lang.StringIndexOutOfBoundsException: Index 39 out of bounds for l fi; od; fi; return r; endx then # convert relators into list of lists ifl1=(g2g); pcgs := G; else Pcgs ) java.lang.StringIndexOutOfBoundsException: Index 7 out of bounds for length 7 r := [else o ; for .()]do r[i] := []; for j andunbind r[ ; od; r[i][i] :java.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for len od # # create collector for Gif [n,-.java.lang.StringIndexOutOfBoundsException: Range [32, GcollAdd ,[i ) +java.lang.StringIndexOutOfBoundsException: Index 25 out of bounds f relatorsreturn r; .orders: o; # create stacks java.lang.StringIndexOutOfBoundsException: Range [16, 9) out of bounds for lengt lestack: []java.lang.StringIndexOutOfBoundsException: Index 23 out of bounds for Gcoll.pstack := []; .cstack: [; Gcoll.mstack := []; # create collector list Gcoll.list =List( pcgso: RelativeOrders(pcgs ) # in case we are not interested in the module[]: [l]+java.lang.StringIndexOutOfBoundsExcep for if w[][l1<c]then # copy collector and add module generatorsi]j]=word(pcgs, pcgs[]java.lang.StringIndexOu r #if r[i][i] := wordpcgs,cgs[^[]) # create module gens (the transposed is a technical detail) # create collector for G =List(.generators,i-ImmutableMatrixl1 : Length(x; r.l2 : Length([]; mzero :ImmutableMatrix(Mfield,Zero(.one); # add avoid r. Gcoll.stack =[]; isSplitjava.lang.StringIndexOutOfBoundsException: Range [20, 19) out of bounds f k:=CharacteristicGcoll.cstack : [; for i in [ 1 .. Length(r.ordersmx[i] : mx[]ml[i]; for j in [ 1 .. i ] do # we can avoid if mx[=[]i]; # was: r.orders[i] <> k and r.orders[j] <> k then AddSetravoidjava.lang.StringIndexOutOfBoundsException: Index 41 out of bounds fo fi; if # copy collector and module generators od; fi; # and return collectormx[]:ml[]java.lang.StringIndexOutOfBoundsException: Index 45 return r; end ); ############################################################################# ## #F AddEquationsSQ( eq, t1, t2 ) ## isSplit local i # if <t1> = <t2> return if for i in[n -1.g1 # compute <t1> - <t2> t1 := ShallowCopyl] :=1java.lang.StringIndexOutOfBoundsException: Index 26 out of b for i in [ 1 .. notl1 avoidthen if IsBound(java.lang.StringIndexOutOfBoundsException: Index 21 out of bounds for if IsBound(t1[i]) then []]=mone; else t1[i] := -t2[i]; fi; fi; od; l : ]java.lang.StringIndexOutOfBoundsException: Index 25 out of bounds for length : ]java.lang.StringIndexOutOfBoundsException: Index 12 out of bounds for length 1 in 1.Length(t1)]do if IsBound(t1[i]) then forjin 1 . eq.imension]do if [i]< eq.vzero then l: l+; ( []i, v); ShrinkRowVector(l[j][i]); fi; od; od; # and reduce lines n: eq.dimension; for j in [1.. n] java.lang.StringIndexOutOfBoundsException: Range [0, 28) out of boun =l]; v := Length(x); if 0 < java.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0 while 0 < v and IsBound(eqjava.lang.StringIndexOutOfBoundsException: Index 0 out of bo c := - t1]: t2[i]; ifrelatorsg]g] >0 then if IsBound(x[i]) then x# make lines AddCoeffs((x[ .system[][,c) ShrinkRowVector(x[i]); if = (x[i] then Unbind( l1 = (g^2+/2java.lang.StringIndexOutOfBoundsException: Index 28 out of boun fi; else x[i] := c * eq.system fi; od v := Length( ; if 0 java.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0 ; if 0 < v then &nsp; in[ ..C do for ineqs java.lang.StringIndexOutOfBoundsException: Index 12 out of bounds for le od # generated p (x] system]i ifk ShrinkRowVector j =1 if 0 = Length(x[i]java.lang.StringIndexOutOfBoundsException: Index 38 out of bounds =[java.lang.StringIndexOutOfBoundsException: Index 12 out of bounds for length 1 fi; x[i] ri :[; fi; od; v := [ : .Ci-); if [od java.lang.StringIndexOutOfBoundsException: Index 11 out of bounds for length 11 0< eqsystem[]: ( Length eqod fi; od; end ############################################################################# ## #F SolutionSQ( C, eq ) ## Gcoll :[; local x, e, d, t, j, v p > # construct null vector n : casewe are not interestedAddCoeffs [j] [] -) iin[1. nrels(avoid) java.lang.StringIndexOutOfBoundsException: Index 51 out of bou java.lang.StringIndexOutOfBoundsException: Range [4, 1) out of bounds for length od;od; # generated position C.unavoidable := []; j := 1 d: .nrelsLength.)] java.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0 if not i in C.avoid then C =; j in java.lang.StringIndexOutOfBoundsException: Index 23 out of bounds for length fi; # blow up vectors t : ]; w := []; # we can avoid for j Add(, ; if =(.) v := ShallowCopy(n); i ineq.pos[] do if not ; p : eq.*(.java.lang.StringIndexOutOfBoundsException: Index 51 out of bounds for le v{ ; ; od; ShrinkRowVector(java.lang.StringIndexOutOfBoundsException: Index 0 out of bounds ############################################################################# AddSet( #F AddEquationsSQ( eq, t1, t2 ) fi; od; # normalize system v := 0*eq.vzero[1]; for i in w do for j in w do if j > i then p:=t[]i; if p <> e := []:=ShallowCopy(t[]) for { java.lang.StringIndexOutOfBoundsException: Index 18 out of bounds for length ShrinkRowVector(t[j]); java.lang.StringIndexOutOfBoundsException: Index 18 out of bounds for length 0 fi; odt1]: -t2java.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for le od # compute homogeneous solution InstallGlobalFunction v := []; e:eq[10java.lang.StringIndexOutOfBoundsException: Index 23 out of bounds for len for i x := ( n =engthG; x[i] := e; for j in w do if j = i x[j] := -t[j][i]; fi; java.lang.StringIndexOutOfBoundsException: Index 11 out of bounds for length 11 Add v, x)java.lang.StringIndexOutOfBoundsException: Index 20 out of bounds for len od; thC)then returnvjava.lang.StringIndexOutOfBoundsException: Index 17 out of bounds for len fi; # construct null vector :[ [ 1 ..els ]do Append(,q. ) od; <v then = (-)n + Length(x[]) ; # construct a blow up matrix i : od; eqnrels ] do if not Append( ,(j-1)java.lang.StringIndexOutOfBoundsException: Index 33 out of bounds f fi; od; # blowup the vectors :[java.lang.StringIndexOutOfBoundsException: Index 12 out of bounds for length 1 forx vjava.lang.StringIndexOutOfBoundsException: Range [4, 1) out of bounds for l d : w1word >w2. then Add( e, d ); od; # and return return e; end ); ############################################################################# ## #F TwoCocyclesSQ( C, G, M ) ## consistency 2: j^(p-1) (ji) = j^p i 2 j^-1 ( ^i local pairs , , k, w1 ,eq,p n; # get number of generators n := Length(end)java.lang.StringIndexOutOfBoundsException: Index 6 out of bounds f # collect equations in <eq>#F SolutionSQ( C, eq ) eq := rec( vzero := C.mzero[1], : mzero, :=java.lang.StringIndexOutOfBoundsException: Index 34 out of bounds for length 2 nrels: n+n/2, spos = [], Append( n.zero )java.lang.StringIndexOutOfBoundsException: Index 30 out of bounds # precalculate (ij) for i > j pairs := List( [1..n], x -> [] ); ] do for j in [ 1 .. i-1 ] od; # consi 3: k ii(-1)=()^ +1 od; # consistency 1: k(ji) = (kj)i for i in [ n, k in for forj[ .(.stem java.lang.StringIndexOutOfBoundsException: Range [44, 45) out of -1 ..j+ java.lang.StringIndexOutOfBoundsException: Range [36, 35) out of bounds fo =for i in eq.spos java.lang.StringIndexOutOfBoundsException: Range [36, 37) out of b p-]); if w1.if w1.word <> w2then (kji" {+.engtheqystem]i)]: system[]i] else (eq w1tail else fi od od; od; # consistency 2: j^(p-1) (ji) = j^p i for i in [ n, n-1 .. :0*qv[1; for j i wdo1 .] do p : C.orders[]java.lang.StringIndexOutOfBoundsException: Index 30 out of bounds fo w1 := CollectedWordSQ( C, [j,p-1], :CollectedWordSQ( C i,, = p1,[1) [i,1] java.lang.StringIndexOutOfBoundsException: Index 35 out of bounds for lengt if w1.word <> w2.word then j(-1)(ji >j^ " ([]java.lang.StringIndexOutOfBoundsException: Index 42 out of bounds else AddEquationsSQ( # and return solution ; od od; # consistency 3: k (i i^(p-1)) = (ki) i^p-1 : for i in [ n, n-1 .. ## p := C.orders[i]; +1 do java.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0 java.lang.StringIndexOutOfBoundsException: Index 27 out of bounds for length 23 =( C,CollectedWordSQ 1 1) p]) if w1.word <> w2.word Error( "k i^p <> (ki [k]]=[]] java.lang.StringIndexOutOfBoundsException: Index 53 out of lse AddEquationsSQ( eq, w1.tailsystem = ]) fi; od; od; # consistency 4: (i i^(p-1)) i = i (i^(p-1) i) for i in [ in .ijava.lang.StringIndexOutOfBoundsException: Index 28 out of bounds for ; [i,1] ); in ,n java.lang.StringIndexOutOfBoundsException: Range [33, 34) out of bounds for # power relator if i = j then .java.lang.StringIndexOutOfBoundsException: Range [37, 34) out of bound m := C.mone; for e in [ 1 .. C.orders[i] ] do R[i][x =(ew>(, od; # conjugate else R[i][x] := R[i][x] - M[j]; R[j]] : R[][] MI[]Mi*[j C ; fi fi; # compute fox derivatives m: Cmone r := C.relators[i][j]; if r <> 0 then in Length()1,(r)- java.lang.StringIndexOutOfBoundsException: Range [9, 8) out of b for e in [ 1 .. r[k+1] ] do R[r[k]][x] := R[r[k]][x] + m; := M[rk] *mjava.lang.StringIndexOutOfBoundsException: Index 41 out of bounds for l ListWithIdenticalEntries(2*dim od; fi; od; od; make one list m := []; r = n*n)/; for i in [ 1 .. n ] do for k in [ 1 .. d ] do = ] j [1 . r]do Append e R[i][][k ) od; Add( m, e ); od; od; # compute a base for <m> (m)java.lang.StringIndexOutOfBoundsException: Index 22 out of bounds for length end ); ############################################################################# ## #F TwoCohomologySQ( C, G, M ) ## InstallGlobalFunction, function(C , java.lang.StringIndexOutOfBoundsException: Inde local cc, cb; cc := TwoCocyclesSQ i in [1..im]do if Length( cc ) > 0 then cb: ( C,G M )java.lang.StringIndexOutOfBoundsException: Index 43 out of bounds for l if Length( C.avoid ) > 0 then cb := java.lang.StringIndexOutOfBoundsException: Index 26 out of bounds for length fi; if Length( cb ) > 0 then cc := gens:=GeneratorsOfGroup; fi; fi; ; end ); ############################################################################# ## #M TwoCocycles( G, M ) ## InstallMethod( TwoCocycles, "generic for in [..im]java.lang.StringIndexOutOfBoundsException: Index 24 out of bo true, [ if Length( : =TwoCoboundariesSQ(C,,M) for in [1. java.lang.StringIndexOutOfBoundsException: Range [28, 29) out of bounds cb := SumIntersectionMat( cc, cb )[2]x:nonone,len,g fi; if Length( cb ) collect the word in factor group cc :C. fi; fi; return cc; end ); ############################################################################# ## #M TwoCocycles( G, M ) ## InstallMethod( TwoCocycles, java.lang.StringIndexOutOfBoundsException: Range [20, 19) out of bounds for leng true, [ IsPcGroup; 0, functionjava.lang.StringIndexOutOfBoundsException: Range [0, 14) out of bounds f local C; C : mapped:=function(l TwoCocyclesSQ C G ; end ); ############################################################################# ## #M TwoCoboundaries( G, M ) ## java.lang.StringIndexOutOfBoundsException: Index 6 out of bounds for length 6 "generic method for pc groups", true, [ IsPcGroup, java.lang.StringIndexOutOfBoundsException: Index 23 out of bounds for 0, function( G,M ) local C; C := CollectorSQ( G, i=java.lang.StringIndexOutOfBoundsException: Index 8 out of bou = [; end ); # non-solvable cohomology based on rewriting ############################################################################# ## #M TwoCohomology( G, M ) ## , "generic method for pc groups", true, [ IsPcGroup, Add( m, e 0, function( G, M ) local C, d, z, co, return Base)java.lang.StringIndexOutOfBoundsException: Index 22 out C := CollectorSQ( G, M, false; d := Length( C.orders );# d := d * (d+1) / 2; ( List( fi; : TwoCocyclesSQ(C, G,M ); od cb := TwoCoboundariesSQ( C, G, M ); cb := SubspaceNC( co, cb ); return rec(group : , module : . 0 then collector isPcCohomology:=true, : cc := BaseSteinitzVectors( cc,# if false then presentation : ; end ); # code for generic 2-cohomology (solvable not required) InstallMethod(################################################################## ## function(G,mo) local field,fp,fpg,gens,hom,mats,fm,mon,tzrules,java.lang.StringIndexOutOfBoundsE len1,l2,java.lang.StringIndexOutOfBoundsException: Index 15 out of bounds for le findtail,colltz, java.lang.StringIndexOutOfBoundsException: Index 1 out of bounds for length 0 nonone,olen,dag; #collect :fp; fm=on.; i, # collect from left# fi; :=(L((java.lang.StringIndexOutOfBoundsException: Index 31 out of bounds for leng ia #does ruleapplyat i? p:RuleAtPosKBDAG(ag,i; if IsInt(p) then java.lang.StringIndexOutOfBoundsException: Index 12 out of bounds for length 12 a{[i+Length(tzrules[p,G ; i:=Maximum(0,i-maljava.lang.StringIndexOutOfBounsp; # collect from left i:=1; whilejava.lang.StringIndexOutOfBoundsException: Range [0, 1) out of bounds for l # does a rule apply at position i? p:=RuleAtPosKBDAG(dag,wrd,i); if () rulpos[]>fail then p:=rulpos[p]; tail forini.Length(ens] wrd:=Concatenation(wrd{[1..i-1 (gi[j) if p in ; if IsIdenticalObj(v[p],zero) then v[p]:=mapped(tail); else p:p+appedjava.lang.StringIndexOutOfBoundsException: Range [36, 34) out of bounds ; ; i:=Maximum(0,i-mal); # earliest which could be affected ; i:=i+1; od; return [wrd,v];:Maximum(1000,*(pS(p,); end; field:=mo.field; simi:=fail; # if false then # # old general KB code, left for debugging # fp:=IsomorphismFpGroup(G); # fpg:=Range(fp); # fm:=IsomorphismFpMonoid(fpg); # mon:=Range(fm); # # if IsBound(mon!.confl) then # tzrules:=mon!.confl; # else # kb:=KnuthBendixRewritingSystem(mon); # MakeConfluent(kb); # tzrules:=kb!.tzrules; # mon!.confl:=tzrules; # fi; # else # new approach with RWS from chief series mon=) fp:=mon.fphom; fpg:=Range(fp); fm:=mon tzrules:=List(RelationsOfFpMonoid(mon),x->List(x,java.lang.StringIndexOutOfBounds # fi; dag:=EmptyKBDAG(Union(List(GeneratorsOfMonoid(FreeMonoidOfFpMonoid(mon)), LetterRepAssocWord))); mal:=Maximum(List(tzrules,x->Length(x[1] sim=IsomorphismSimplifiedFpGroup() for i in =) AddRuleKBDAG(dag,tzrules[i][1],i); od; gensjava.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length x>PreImagesRepresentative,x); hom:=GroupHomomorphismByImagesNC(G,Group(mo.generators), GeneratorsOfGroup(G (gensx-ImagesRepresentativehom),.field # new gens l1:=GeneratorsOfGroup(fpg); l1:=Concatenation(l1,List(l1,Inverse)); formalinverse:=[]; for i in l1 do AssocWord(nderlyingElementImagesRepresentativefm,)); o:=LetterRepAssocWord(UnderlyingElement(ImagesRepresentative( ListWithIdenticalEn if Length(j)<>1 or Length(o)<m:=p; formalinverse[j[1]]:=o[1]; od; # rules that describe formal inverses, or delete generators of order 2 get no :=FilteredDerivedSeriesOfGroup() hastail:=[]; rules:=[]; genkill:=];# relations that kill generators. Needed for presenation.# roughly 5 for 1000, 30 for r in # allow to enfo test coverage if Length(r[1])>=2 then Add(,r) if Length(r[1])>2 or (Length(r[1])=2 and (Length(r[2])>0 or formalinverse[r[1][1]]<>r[1][2])) then ; fi; else generator is not used m:=First(RelationsOfFpMonoid(mon),x->List(x,LetterRepAssocWord)=r); m:=List(m,x->PreImagesRepresentative(fm,ElementOfFpMonoid(FamilyObj(One(mon)),x)) :List(nderlyingElement) # free group elements/words if not IsOne(m[1]*Subword(m[2],1,1)) then akea generatorredundant (byexpressing it the # other gens)? If so, remember relation as needed to kill this generator, butifit=fail then # no influence on Cohomology calculation (which just uses the rest) m1 in GeneratorsOfGroup(FreeGroupOfFpGroupF()! then Add(genkill,r); fi; fi; fi; od; rulpos:=List :pjava.lang.StringIndexOutOfBoundsException: Index 19 out of bounds htpos:=List([1..Length# avoid working hard for outer automorphisms model:=ValueOption("model e:=ilteredDerivedSeriesOfGroupm, if model<>fail then q:=GQuotients(model,G)[1]; pre:=List(gens,x->PreImagesRepresentative(java.lang.StringIndexOutOfBoundsExcepti ker:=KernelOfMultiplicativeGeneralMapping(q); m:Laste) l1:=GModuleByMats(LinearActionLayer(java.lang.StringIndexOutOfBoundsException: I MTXIsIrreducible)java.lang.StringIndexOutOfBoundsException: Index 26 out of boun miso:=MTX.Isomorphismfi ,x)); pcgs:=PcgsByPcSequence(FamilyObj(One(model)),new); vector solvec:=[]; one:=Onefalse<.field m:=GroupGeneralMappingByImagesNC(fpg,model,GeneratorsOfGroup( m, mats:=List(GeneratorsOfMonoid(mon)x>java.lang.StringIndexOutOfBoundsException: Ra x->ImagesRepresentative(m, PreImagesRepresentative(fm,x))); #Elements for monoid generators nonone:= (,, I()java.lang.StringIndexOutOfBoundsException: Index 60 out of bounds pre:=mats; onemat:=One(G); for r ((p))java.lang.StringIndexOutOfBoundsException: Index 59 out of bounds for leng m:=LeftQuotient; m: 50 solvec:Concatenationsolvec,*ne(); od; fi; onemat:=One(mo.generators[1]); zerovec:=java.lang.StringIndexOutOfBoundsException: Index 13 out of bounds for l mats:= wasbold= x>magesRepresentative,(fp, PreImagesRepresentative(fm,x)))); # matrices for monoid generators one:=One(mats[1 ->notIsOne(mats[]); zero:=zerovec; dim:=Length(zero # Rewriting produces a bad presentation. Rather rebuild a new nvars:=dim*Length(hastailjava.lang.StringIndexOutOfBoundsException: Index 73 out #rk:=0; es(nvars,()java.lang.StringIndexOutOfBoundsException: Index 76 out of bounds for eqs:=MutableBasisx>(FamilyObjOne(ps),) olen: = (,,hom is)fijava.lang.StringIndexOutOfBoundsException: Index 68 out of boun for :=List(x-ImagesRepresentative,) trule ,i," BasisVectors(eqs), equations)java.lang.StringIndexOutOfBoundsException: 50 len1:=Length(l1); forj [.Length() do l2:=rules[j][1]; m: for o in [1..m-1] do # possible overlap Length start:=len1-o; if ForAll([1..o],k->l1[start+k]=l2[k]) then # apply l1 first new:else c:=collectail(new); v1:=c[2];c:=c[1]; if fi; v1[i]:= [i][1 java.lang.StringIndexOutOfBoundsException: Index 21 out of bounds for length 21 forin[.Length()do l2:java.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0 m:=Minimum(len1,Length(l2)); for o for i 1.engthw]o start:=len1-o; if ForAll([1..o],k->l1[start+k]=l2[k]) :java.lang.StringIndexOutOfBoundsException: # apply l1 first :GModuleByMats(nearActionLayerGroupp)cgs).eld); od IsIrreduciblem; v1=[]c:[]java.lang.StringIndexOutOfBoundsException: Index 27 out of bounds for l if i in hastail then v1i]=v1[i]+mapped(l2{[o+1..Length(l2]); fi; # apply l2 first new:=Concatenation(l1{[1..len1-o]},rules[j][2]); v2:=collectail(new); c:[i,ormalinversemats:(GeneratorsOfMonoidmon, if c<>v2[1] then Error("different in factor"); #else Print("Both reduce to ",c,"\n"); fi; :v2[2; if =matsjava.lang.StringIndexOutOfBoundsException: Index 14 out of bounds for len [j]=[]ne; fi; if :indtail([2,rjava.lang.StringIndexOutOfBoundsException: Index 40 out of bounds c:= od; for in java.lang.StringIndexOutOfBoundsException: Index 31 out of bounds for lengt ifk>2[ then Addbds,ew; r fijava.lang.StringIndexOutOfBoundsException: Index 5 out of bounds for length 5 l [..im do zerovec=(emat]; []r}=ew[l] fi; ; ; for k cjava.lang.StringIndexOutOfBoundsException: Index 25 out of bounds for lengt ifmodel<and IsZero(lvec*then ("odel does not fit"); fi; #AddSet(eqs,ImmutableVector(field,k)); #k:=SiftedVector(eqs,ImmutableVector(field,k));=(FreeGroupOfFpGroupf); #Add(alleeqs,ImmutableVector(field,k)); k:SiftedVector,)java.lang.StringIndexOutOfBoundsException: Index 37 out of bound if not IsZero(#rk:=0; CloseMutableBasis(eqs,ImmutableVector(field,k)) local ai fi ; fi java.lang.StringIndexOutOfBoundsException: Index 7 out of bounds for length 7 fi; od; od; olenLength(BasisVectorseqs)java.lang.StringIndexOutOfBoundsException: Index 40 o if(olen) # if twice stayed the same after increaseo [.-]do # possible overlap Length olen[ength( killrelators:=List-LeftQuotientapped2i],(2), ForAll1.o]k-l1[tart+] # position of relators with tails in t k:=List(BasisVectors(eqs),ShallowCopy);; TriangulizeMat() eqs:MutableBasis(ield, List(k,x->ImmutableVector(field,x)),zeroq); fi; hastail od; #eqs:=Filtered(TriangulizedMat(eqs),x->not IsZero(x)); eqs:=ShallowCopy(java.lang.StringIndexOutOfBoundsException: Index 31 out of boun if Length(eqs)=0 then eqs=IdentityMatLength(f) else eqs:=ImmutableMatrix(java.lang.StringIndexOutOfBoundsException: Index 29 out of eqs:((qs);# basis of cocycles fi; Now get Coboundaries # different collection function: Sum up changes for generator i findtail:=function(wrd,nums,vecs) local a,i,j,p,v; a:zerovec; o p:=Position(nums,wrd[i]); if p<>fail then :vecs[] for j in [i+1..Length(wrd)] do v:=v*mats[wrd[j]]; od; a=a+v; fi od; ajava.lang.StringIndexOutOfBoundsException: Index 13 out of bounds for length 13 end; bds:=[]; for i [.Length(ats]do if formalinverse[i] .colltz:=olltz; c:=[i,formalinverse[i]]; for k in one do r:=[k,-k*mats[formalinverse[i] colltz(formalinverse{eversed(wrd)), new:=[]; for j in hastail do v1:=findtail(rules[j][1],c,r); v2:=r.pairact:=function(zy, if notIsZero() n,-) java.lang.StringIndexOutOfBoundsException: Range [0, 10) out of bounds for lengt new:= (,od Add(bds,new od; fi; od; bds:=Filtered(TriangulizedMat(bds),x->not bds: local ; ifgens>java.lang.StringIndexOutOfBoundsException: Range [29, 28) out of bounds fo G:=GroupWithGenerators(gens); fi; r:=rec(group:=G,module:=mo,cocycles:=eqs,coboundaries:a L()<olen[ength()3 prime:Size(ield); a:ImagesRepresentativefm)java.lang.StringIndexOutOfBoundsException: Index 38 out r.:new; new:=[]; one:=One(FreeGroupOfFpGroup(fpg)); k:=java.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0 eqs:=hallowCopy(asisVectorseqs); to monoid word localvjava.lang.StringIndexOutOfBoundsException: Range [14, 13) out of bounds fo local a,i; a:=one; for i in list do a:=a*k[i]; od; return a; end; for i in hastail do # rule that would get the tail Add(,(mapped2rulesi[1],apped2(ulesi[])) od; .presentation i1Length(d) java.lang.StringIndexOutOfBoundsException: Range [32, 33 # relators to kill superfluous generators java.lang.StringIndexOutOfBoundsException: Range [8, 1) out of bounds for length # position of relators with tails in tzrules monrulpos:=List(rules{hastail},x-> ; prewords:=List(ogens,x->UnderlyingElement(ImagesRepresentative(fp,x)))); # normalform word and collect the tails ajava.lang.StringIndexOutOfBoundsException: Ind r.:(,) local v,i,j,p,w,tail; v:=zerovec; # i:1; whilei<(wrd) =java.lang.StringIndexOutOfBoundsException: Index 23 out of bounds fo # does a rule apply at position i? p=d,rdi)java.lang.StringIndexOutOfBoundsException: Index 35 out of bounds for le if :findtail(ulesj]1 p:rulpos[] tail:=wrd{[i+Length(rules[p end wrd:=java.lang.StringIndexOutOfBoundsException: Index 14 out of bounds for lengt if p in hastail then w:= # images of for j in tail do w:w*[j] od; v:=v+w; fi; i:=Maximum(0,i-mal); # earliest which could be affected fi; i: r:rec(roup=G,odule:mo, :left] od; ,v; end; java.lang.StringIndexOutOfBoundsException: Range [0, 3) out of bounds for length monhomfm; r.colltz:=colltz; # inverses of generators r.myinvers:=function(wrd,zy) return [colltz# ext:=FpGroupCocycle(r,zy,true); -r mapped2=mage(om) 2]java.lang.StringIndexOutOfBoundsException: Index 11 out of bounds for length 1 end; r.pairact:=function(zy,pair) local autom,mat,i,imagemonwords,imgwrd,left,right,extima autom:=InverseGeneralMapping(pair[# od; # cache words for images of monoid generators2*x-1))); # new generators if # exte:=v; imagemonwords:autom!imagemonwords imgwrd:=function(nr) local a; if not IsBound(# v:=-ExponentsOfPcElement(epcgs,left)*One(mo.field); # apply automorphism # normalform word and collect the tails a:PreImagesRepresentative,a; a:=ImagesRepresentative v,,,,,ail; a=(, :zerovec a:=ImagesRepresentative(fm,a); a:=LetterRepAssocWord(UnderlyingElement(a)); a:=colltz(a); imagemonwords[nr]:=a; fi; [] end; # normalform word and collect the tails collectail:=function(wrd) local v,i,j,p,w,tail; v:=zerovec; # collect from left i:=1; while i<=Length(wrd) do # does a rule apply at position i? p:=RuleAtPosKBDAG(dag,wrdod; if IsInt(p) and rulpos[p]<> odjava.lang.StringIndexOutOfBoundsException: Index 13 out p:=rulpos[p]; tail:= :Maximum(imal;# earliest which could be affected wrd:=Concatenation(wrd{[1..i-1]},rules[p][2],tailjava.lang.StringIndexOutOfBound if p in hastail then =dim forj intaildo odjava.lang.StringIndexOutOfBoundsException: Index 7 out of bounds for length 7 od; v:=v+w; fi; i:=Maximum(0,i-mal); # earliest which could be affected) fi; :i+; od; [,vjava.lang.StringIndexOutOfBoundsException: Index 21 out of bounds for length end; autom=(pair[]; local dogoimmediately subgroupindexup4java.lang.StringIndexOutOfBoundsException: I =] v:=zerovec; imagemonwords=utom.; v:=v*imgwrd:=function) locala :vw]+2; w:w[] od; return [w,]; end; bas=(mo) new:[; # inverses of generatorsUnderlyingElement(a))java.lang.StringIndexOutOfBoundsExcept : r=java.lang.StringIndexOutOfBoundsException: Index 29 out of bounds for length -return[; # images of generators extim:=List r=g=java.lang.StringIndexOutOfBoundsException: Index 23 out of bounds # collect inverses and interleave generators/inverses java.lang.StringIndexOutOfBoundsException: Index 36 out of bounds for length 36 # evaluate rules in images left=prdout({rules[hastail # invert left v:myinversl[) left:=[v[1],v[2]-left[2]/mapped(left[1])]; ]}); # quotient left^-1*right a< then :prdoutConcatenation(left][ight); if left[1]<>[ebas:((d,() (new,left[2]*mat); od; # debugging code: Construct group and work there # new2:=[]; # ext:=FpGroupCocycle(r,zy,true); # hom:=IsomorphismPermGroup(ext); # ext:=Image(hom); # epcgs:=PcgsByPcSequence(FamilyObj(One(ext)), # GeneratorsOfGroup(ext){[Length(mats)/2+1..Length(GeneratorsOfGroup(ext))]}); # # exta:=Concatenation(List([1..Length(mats)/2],x->[ext.(x),ext.(x)^-1])); # exte:=exta; # # myprd:=function(wrd) # local i,w; # w:=One(ext); # for i in wrd do # w:=w*exte[i]; # od; # return w; # end; # # v:=List([1..Length(mats)/2],x->myprd(imgwrd(2*x-1))); # new generators # v:=Concatenation(List([1..Length(mats)/2],x->[v[x],v[x]^-1])); # exte:=v; # # for i in [1..Length(hastail)] do # left:=rules[hastail[i]][1]; # right:=rules[hastail[i]][2]; # left:=myprd(left)^-1*myprd(right); # v:=-ExponentsOfPcElement(epcgs,left)*One(mo.field); # # Append(new2,v*mat); # od; # if new2<>new then Error("wrong ",pair);fi; # if IsOne(pair[1]) and IsOne(pair[2]) then # if new<>zy then Error("one");else Print("onegood\n");fi; # fi; ImmutableVector,ew) endjava.lang.StringIndexOutOfBoundsException: Index 6 out of bounds for length 6 ; ); ,function() local e,one,m,r,a,c,is,i; e:=[]; one:=One(mats[1); for m in mats do r:=Set(RootsOfUPol(field,CharacteristicPolynomial(m))); a:=List(r,x->java.lang.StringIndexOutOfBoundsException: Index 29 out of bounds fo if Length(a)= bas:bas{1]; Add(,) od; for is:c1; for i in [2..Length(c)] do is:=SumIntersectionMat(is,c[i])[2]a=bas[i]]*.enerators; od;Appendn,left[*) if Length(is)java.lang.StringIndexOutOfBoundsException: Index 19 out of bounds fo returnis; fi; od; return false; end ); BindGlobal("WreathElm",function(b,l,m) local n,ran,r,d,p,i,j; n:=Length(l); ran:=[1..b]# r:=0; d:=[]; p:=[]; # base bit ]; if(is)0 then return is; fi; od; return false fi; end BindGlobal("WreathElm",function(b,l,m) local n,ran,r,d,p,i,j; n:=Length(l); ran:=[1..b]; r:=0; d:=[]; p:=[]; # base bit forw:zy{*(htpos for j in ran do p[r+j]:=r+j^l[i]; od; Add(d,ran+r); r:=r+b; od; # permuter bit.yinvers=functionwrdz) p:=return[(formalinverse{Reversedwrd}, return p; end; BindGlobal("PermrepSemidirectModule",function(G,module) local hom,mats,m,i,j,mo,bas,a,l,ugens,gi,r,cy,act,k,it,p; if not MTX.IsIrreducible(module) then Error("reducible");fi; p:=Size(module.field); if not IsPrime(p) then Error("must be over prime field");fi; k:=Length(module.generators); hom:=GroupHomomorphismByImagesNC(G,Group(module.generators), GeneratorsOfGroup(G),module.generators); ow do immediatelyto normal of up to 4. # This reduces search space it:= w:[] repeat m:=NextIterator(it); if Index(G,m)*p>p^module.dimension then Info(InfoExtReps,2,"Index reached ",Index(G,m), ", write out module action"); # alternative, boring version, mo:=AbelianGroup(ListWithIdenticalEntries(module.dimensiona=ImagesRepresentative :=Pcgsmo; act:=List(module =]java.lang.StringIndexOutOfBoundsException: Index 12 out of bou GroupHomomorphismByImagesNC(mo,mo,bas, List(m,x->PcElementByExponents(bas,x)):noassert)); Assert(3,ForAll(act,IsBijectivemyinversimagemonwords[:a fi imagemonwords[]java.lang.StringIndexOutOfBoundsException: Index 32 out of bounds gi:=SemidirectProduct(G,GroupHomomorphismByImages(G,a, GeneratorsOfGroup(G),act),mo); :rec(roup:gi, ggens:=List(GeneratorsOfGroup(G),x-> ImagesRepresentative(Embedding(gi,1),x)), #vector:=ImagesRepresentative(Embedding(gi,2),bas[1]), basis:=List(bas,x->ImagesRepresentative(Embedding(gi,2),x))); Assert(0,Size(gi)= iin 1.Length(hastail)]do return r; :extim p:RuleAtPosKBDAGdagw,i); else Info(InfoExtReps,2=(eft[]; torsOfGroup)java.lang.StringIndexOutOfBoundsException: Index 38 out of bounds fo x->TransposedMat( right:=prdout(extim{rules[hastail[i]][2 java.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0 ifa<false then # quotient module # basis: supplemental vector and submodule basis :BaseSteinitzVectorsIdentityMatmodule.imension,F()) NullspaceMat(TransposedMat(a{[1]}))); bas:Append-java.lang.StringIndexOutOfBoundsException: Range [23, 22) out of boun # assume we have a generating set for the SDP consisting of the code group work there java.lang.StringIndexOutOfBoundsException: Index 13 out of bounds for length 13 cy:=CyclicGroup(IsPermGroup,p).1; # p-cycle ugens:=[]; # also transversal chosen in complement r:=RightTransversal(G,m); act:=ActionHomomorphism# ; #l:=ListWithIdenticalEntries(Index(G,m),()); for gi in [1..k] do l:=[]; for j in [1..Length(r)] do # matrix for Schreier gen a:=ImagesRepresentative(hom, r[j]*G.(gi)/r[PositionCanonical(r,r[j]*G.(gi))]); # how does it act on bas[1]-span, get factor a:=Int(SolutionMat(bas,bas[1]*a)[1]); # write down permutation that acts as mult by a if IsZero(a) then l[j]:=(); else l[j]:=MappingPermListList([1. w: :=oncatenation(List1.Lengthm)2,-v],[]^-]) fi; od; Add(ugens,WreathElm(p,l,act[gi]) )] od; # module generatorjava.lang.StringIndexOutOfBoundsException: Index 28 out of bound for j in [1..Length(r)] do #r[j]*g/r[j^g]); Note j^g=j here as kernel of permrep# if new2<>new then Error("wrong ",pair); lj]: cy# if new<>zy then Error("one");else Print("onegood\n");fi; # fi; Add(ugensjava.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for le gi:=Group; Assert(0, java.lang.StringIndexOutOfBoundsException: Index 57 out of bounds for l if returnr a:end; if NrMovedPoints gi:=mageBindGlobal(MatricesStabilizerOneDim" (ield,java.lang.StringIndexOutOfBoun ugens:=List(ugens,x->ImagesRepresentative(a,x)); fi] fi; r=(groupgi,:ugens{[.k}) # compute basis :[1}java.lang.StringIndexOutOfBoundsException: Index 22 out of bounds for length gi:=ugens{[1..k]}; ugens:=ugens{[k+1]}; i:=1; while Length(bas)<module.dimension do for in[..k]do :i*odulegenerators[j]; if SolutionMat(bas,a)=fail then Add(bas,a); # debugging code: Construct group and fi; od; i:=i+1; od; # convert back to standard basis r.basis:=List(Inverse(bas),x->LinearCombinationPcgs(java.lang.StringIndexOutOfBou return r; fi; fi; until false; end); BindGlobal("java.lang.StringIndexOutOfBoundsException: Index 13 out of bounds fo local best,u,v,start,cnt,runtime; runtime:=GET_TIMER_FROM_ReproducibleBehaviour# start:=runtime(); best:# v:=Concatenation(List([1..Length(mats)/2],x->[v[x],v[x]^-1])); cnt:=0; while runtime()-start<time or cnt<100 do cnt:=nt+1; u:=TrivialSubgroup(g); repeat# right:=rules[hastail[i]][2]; v:=u; u:=ClosureGroup(u,Random(g)); until IsSubset(u,n); if IndexNC(g,v)<IndexNC(g,best) then best:=v;fi; od; return best; end) p:p/(d,)) if IsOne(pair[1]) and IsOne(pair[2]) then InstallGlobalFunction(FpGroupCocycle,function(arg) local); it,hom,trysy,prime,mindeg,fps,eiBindGlobal",(m) tab,end; mon,ord,mn,melmvec,killgens,frew,fffam,ofgens,rws,formalinverse; # function to evaluate product (as integer list) in gens (and their # inverses invs) with corresponding action mats evalprod:if IsPrimep) java.lang.StringIndexOutOfBoundsException: Index 22 out of b local new,i; new:=[[],zerob]; for i in w do if i>0 then collect:=r.tailforword(Concatenation(new[1],gens[i][1]),z); new:=[collect[1] collect:=r.tailforword(Concatenation new=[]2+ew2][i]invs-[java.lang.StringIndexOutOfBoundsException: Index 1 out of b fi; odjava.lang.StringIndexOutOfBoundsException: Index 10 out of bounds for length 9 return new; ; java.lang.StringIndexOutOfBoundsException: Range [6, 5) out of bounds for length local i,a,w; w:=One(f); for i in [1..Length(v)] do a:=Int(v[i]); if prime=2 or a*2<prime then w:=w*gens[2*i-1+mn]^a; else a:prime-a; w:=w*gens[2*i+mn]^a; fi; od; return w end; # formal inverse of module element formalinverse:=function(w) local l,i; :[; for i in Reversed(LetterRepAssocWord(w)) do if IsEvenInt(i) then Add(l,i-1); else Add(l,i+1);fi; od; return java.lang.StringIndexOutOfBoundsException: Index 23 out of bounds for leng end; java.lang.StringIndexOutOfBoundsException: Range [6, 1) out of bounds for length z ogensgroup; =.dimension prime:=Size(r.module.field); if ValueOption("normalform")<> :Sizefield; mon:=fail; else k=Length(module.enerators) # will allow for reduced multiplication. Do so before the group, so there is no:GGroup.generat # issue about overwriting variables that might be needed later mon:=Image(r.monhom); ofgens:=FreeGeneratorsOfFpMonoid(mon); fffam:=FamilyObj(One(FreeMonoidOfFpMonoid(mon))); frew:educedConfluentRewritingSystem(mon); # generators that get killed. Do not use in RHS (TransposedMat(a[1}) ReducedForm(frew :NextIterator( bas:java.lang.StringIndexOutOfBoundsException: Ran killgens:=Union(List(killgens,x->LetterRepAssocWord(UnderlyingElement(x)))); str:=List(GeneratorsOfMonoid(mon),String); mn:=Length(str); for i in [1..dim] do Add(str,Concatenation("m",String(i))); Add(str,Concatenation("m",String(i),"^-1")); od f:=FreeMonoid(str); gens:=GeneratorsOfMonoid(f); rels:=[]; # inverse rules for i in [1,3..Length(gens)-1] do Addg[][+]Onef]; Add(rels,[gens[i+1]*gens[i],One(f) =] od; # module is elementary abelian inm+1,n3.Lengthstr)1 do if prime=2 then Add ListG,> Add(rels,[ImagesRepresentati(java.lang.StringIndexOutOfBoundsException: Range [5 else if ()then Add(rels,[gens[i]^j,gens[i+1]^(j-1)]); Add(rels,[gens[i+1]^j,gens[i]^( else fi; for j in [i+2..Length(str)] do Add(rels,[gens[j]*gens[i],gens[i]*gens[java.lang.StringIndexOutOfBoundsException Add(rels,[gens[j]*gens[i+1],gens[i+1]*gens[j]]); od; od; # module rules for i in [1..mn] do ifnot injava.lang.StringIndexOutOfBoundsException: Index 31 out of bounds for len for j in [mn+1..Length(str)ifa>falsethen new:=r.module.generators[QuoInt(i+1,2)]; if IsEvenInt(i) then new:=new^-1; fi; new:=new[QuoInt(j-mn+1,2)]; if IsEvenInt(j) then new:=-new;fi; Add(rels,[gens[j]*gens[i],gens[i]*melmvec(new)]); ; fi od # rules with tails for i in [1..Length(r.presentation. cy:=CyclicGroup(IsPermGroup,p).1; # p(IsPermGroupp java.lang.StringIndexOutOfBoundsException: Range [31, 30) out of bounds for leng new:=List(new,x-> [:Filtered 1.nrels >xt) ) 2:[]elmvec{i*+.idim}; Add(rels od; java.lang.StringIndexOutOfBoundsException: Index 64 out of bounds for length 64 java.lang.StringIndexOutOfBoundsException: Range [34, 6) out of bounds for lengt for =; new=(fffam[]java.lang.StringIndexOutOfBoundsException: Index 45 out of bounds fo new:=[new =] ,1.}); v=(n); for i .sposj] do ; fi; java.lang.StringIndexOutOfBoundsException: Range [0, 27) out of bounds for le # normalize system ord:=OrderingOfRewritingSystem(ReducedConfluentRewritingSystem(mon) forj inw do:() if HasLevelsOfGenerators(ord) then ord:=WreathProductOrdering(f, tj []- ; ListWithIdenticalEntries(2dim,1))) else ord:= Concatenation(ListWithIdenticalEntries(mn,2), ListWithIdenticalEntries(2 fi; :WreathProductOrdering(f, ifj> then ListWithIdenticalEntries(2*dim,1))); fi; # if there is any [x^2,1] relation, this means the inverse needs to be # mapped to the generator. (This will have been skipped in the rulpos) for i if Length(i[2])=0 and Length(i[1 ; and Length(Set(LetterRepAssocWord(: [ new:=LetterRepAssocWord(# and return IsOddIntnew) Add(rels,[InstallGlobalFunction( TwoCocyclesSQ on:; fi; fi; od; # temporary build to be able to reduce :FactorFreeMonoidByRelations(f,rels); rws=((ne(on),djava.lang.StringIndexOutOfBoundsException: Index 61 out of bounds # handle inverses that get killed for i in killgens do new od; fori in [ n n1. 1] do new:=MappedWord(new for k in n,n- . j+1 ]java.lang.StringIndexOutOfBoundsException: Ind # now left multiply with non-inverse generator od; j:=ReducedForm(rws,formalinversefori [, n1 .1 do Add(rels,[gens[i], (,[[i*[+]Onep := Cordersj; od; mon(f,els; rws:=KnuthBendixRewritingSystem( java.lang.StringIndexOutOfBoundsException: Range ReduceRules(rws); MakeConfluent(rws); # will add rules to kill inverses, if needed SetReducedConfluentRewritingSystem(mon,java.lang.StringIndexOutOfBoundsException fi[[x =jx]+j*[]M]- ; =.; :O((mon); zerob:= for k[r-,Lengthr ifHasLevelsOfGenerators(ord) then ListWithIdenticalEntries(dim,Zero(r.java.lang.StringIndexOutOfBoundsException: R n:=Length(ogens); gensmrep:=List(java.lang.StringIndexOutOfBoundsException: Range [12, 1) out of b x->[LetterRepAssocWord(UnderlyingElement(e: []; (e,Rij[]; # module generators Append(gensmrep,List(IdentityMat(r.modulereturn BaseMat; x->[java.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0 invsmrep## forin[1.]java.lang.StringIndexOutOfBoundsException: Index 22 out of bounds for l Add(str,Concatenation("m",String(i))); od; f=() (f) rels:=[]; for i in [1..Length(r.presentation.relators) #M TwoCocycles( G, M ) j in [.im do new:=new* java.lang.StringIndexOutOfBoundsException: Index 42 out of bounds for l od Add(rels,new); od; in [1.(.presentation)do method pcjava.lang.StringIndexOutOfBoundsException: Index 35 out of bounds for le Add(java.lang.StringIndexOutOfBoundsException: Range [9, 8) out of bounds for le od; for j:ReducedForm Add(rels,gens#M TwoCohomology( G, M ) for j in [i+1..Lengthjava.lang.StringIndexOutOfBoundsException: Index 0 out of boun ,(gens,ens)java.lang.StringIndexOutOfBoundsException: Index 38 out of bounds for od; for i := SubspaceNC,cb; forj in [.dim java.lang.StringIndexOutOfBoundsException: Range [24, 25) out of boun Add(rels,gens[n+j]^gens[ str:List, LinearCombinationPcgs(gens{[n+1..Length(gens)]},r.module.generators[endjava.lang odjava.lang.StringIndexOutOfBoundsException: Index 8 out of bounds for length 7 ; frelsjava.lang.StringIndexOutOfBoundsException: Index 13 out of bounds for lengt SetSize(fp,Size(r.group)*prime^r.module.dimension); java.lang.StringIndexOutOfBoundsException: Range [12, 6) out of bounds for lengt wasboldjava.lang.StringIndexOutOfBoundsException: Index 21 out of bounds for len :functiona rels:=MakeFpGroupToMonoidHomType1 java.lang.StringIndexOutOfBoundsException: Rang SetReducedMultiplication(fp); fip=(,)java.lang.StringIndexOutOfBoundsException: Index 33 out of bounds for len if Length(arg)>2 and arg[3]=true then if IsZero(z) and MTX.IsIrreducible(r.module) then # make SDP directly m:=PermrepSemidirectModule iin[.Length {+(rulesp[]java.lang.StringIndexOutOfBounds p:= ; # test is cheap here, thus no NC :GroupHomomorphismByImages(,pjava.lang.StringIndexOutOfBoundsException: Range [6 a:=a*; else #sim:=IsomorphismSimplifiedFpGroup(fp); r,x->zerojava.lang.StringIndexOutOfBoundsException: Index 27 out of bounds for le fps:=Image(sim, java.lang.StringIndexOutOfBoundsException: Index 0 out of bounds g:=java.lang.StringIndexOutOfBoundsException: Range [0, 10) out of bounds for le quot: j +ens java.lang.StringIndexOutOfBoundsException: Range [35, 36) out of bound *GroupHomomorphismByImages(fp,g,GeneratorsOfGroup(fp), java.lang.StringIndexOutOfBoundsException: Index 20 out of bounds for length 5 v[]:v[]+apped(tail); hom: fijava.lang.StringIndexOutOfBoundsException: Index 13 out of bounds for leng GeneratorsOfGroup(r.group),r. LinearCombinationPcgs(gens{[n+1(gens)],r.[i]j]) fi; p:=Image(quot); trysy:(100050IndexNC,ylowSubgroup(p,prime))java.lang.StringIndexOutOfBoundsExcep # allow to enforce test coverage if ValueOption("forcetest")=trueogens java.lang.StringIndexOutOfBoundsException: R mindeg# fm:=IsomorphismFpMonoid(fpg); if# if IsBound(mon!.confl) then java.lang.StringIndexOutOfBoundsException: Index 13 out of bounds for length 9 # mon!:tzrules; it:fail; Size()SizefifIsZero( =(; # we allow do go immediately to normal subgroup of index up to 4. # if # re-use the first quotient, helps with repeated subgroups iterator if p=r.group then m:=r.group; j:=LetterRep(((,i))java.lang.StringIndexOutOfBoundsException: Index 73 out of bo :p fi; # avoid working hard for outer automorphisms e=(m)java.lang.StringIndexOutOfBoundsException: Index 48 out of bounds for lengt java.lang.StringIndexOutOfBoundsException: Index 61 out of bounds for length 61 x->IndexNC(m,x)^4<=Size(m)); m:=Last(e); it:=DescSubgroupIterator(m:skip:=LogInt( mindeg:2; fijava.lang.StringIndexOutOfBoundsException: Index 13 out of bounds for length 1 wasbold:=false; m:=NextIterator(it); # catch case of large permdegree, try naive first if Index(p,m)=1 and IsPermGroup( mListUjava.lang.StringIndexOutOfBoundsException: Ra and NrMovedPoints in # take set stabilizer of orbit points. e:=Set(List(if[] GeneratorsOfGroup((amilyObjfpg! m:=Stabilizer(p,e,OnSets); if IndexNC(p,m)>10*NrMovedPoints(java.lang.StringIndexOutOfBoundsException: Range java.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0 fi; if IndexNC(p,m)>10*NrMovedPoints(p) then m:=p; # after all.. wasbold:=false; else wasbold:=true; fi; fi; Info(InfoExtReps,3,"Found index ", new:=List(miso,x->PcElementByExponents(pcgs e:=fail; if Index(p,m)>=java.lang.StringIndexOutOfBoundsException: Index 31 out of bounds f <MatricesStabilizerOneDim(r.module.field, List(GeneratorsOfGroup(m)java.lang.StringIndexOutOfBoundsException: Index 40 out -TransposedMat(ImagesRepresentative(hom,x))^-1))) then InfoInfoExtReps,2"Attemptindex ",ndexp,m); =alsethenInfoIjava.lang.StringIndexOutOfBoundsException: Range [47, 46) out of b ::aximalSubgroupClassReps; # up to index 50 # the rewriting seems to be sufficiently spiffy that we don't # need to worry about this more involved process. # this might fail if generators are killed by rewriting x-I(homPreImagesRepresentativefpjava.lang.StringIndexOutOfBoundsException: Index n # fp group using rewriting rules, finds its abelianization, # and then lift that quotient sub:=PreImage(quot,m); tab0:=AugmentedCosetTableInWholeGroup(sub); # primary generators mwrd:=List(tab0!.zeroq:=ImmutableVector(field,ListWithIdenticalEntriZero(ield)); ->ElementOfFpGroup(fps),); Info(InfoExtReps,4,"mwrd=",mwrd); mgens=Listmwrd,>(quot,)) nn:=Length ",",Length(asisVectors)""; if IsPermGroup(m) then e:=SmallerDegreePermutationRepresentation(m); mfpi:=IsomorphismFpGroupByGenerators(java.lang.StringIndexOutOfBoundsException: List(mgens,x->ImagesRepresentative(e,x)):cheap); mfpi:e*fpijava.lang.StringIndexOutOfBoundsException: Index 29 out of bounds for mfpi:=IsomorphismFpGroupByGenerators(m,mgens:cheap); ; mmats:List(mgens,-ImagesRepresentative(homx)) :java.lang.StringIndexOutOfBoundsException: Index 27 out of bounds for length 26 ListWithIdenticalEntries(r.module.dimension, c>v21 e:=List(mwrd, x-> :=v22; gensmrep,invsmrep,i)); :=ist(,java.lang.StringIndexOutOfBoundsException: Index 37 out of bounds for len local i; i:=r.myinvers(x[1],z); return [i[1],i[2]-x[2 forinhastail java.lang.StringIndexOutOfBoundsException: Index end); =(htpos[k]-1)*dim+[1..dim]; newfree:=FreeGroup(nn+dim); gensc[l]{}=ew[l; rels:=[]; relations for i in [1..dim] od; Add(rels,gens[nn+i]^prime); for j in [1..i-1] do Add(rels,Comm(gens[nn+i],gens[nn+#k:SiftedVectorImmutableVectorfieldk); od; for in[..Length(gens)] if not ()then rels,ens[nn+]^gens =(,function() LinearCombinationPcgs(gens{[nn+1.java.lang.StringIndexOutOfBoundsException: Inde od; od; #extended presentation for i in RelatorsOfFpGroup(Range(mfpi)) do i:=LetterRepAssocWord(i); str:=evalprod(i,e,ei,mmats); if Length(str[1])>0 then Error("inconsistent");fi; Add(rels,AssocWordByLetterRep(FamilyObj(One(newfree)),java.lang.StringIndexOutOf java.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0 od; newfree:=newfree/rels; #eqs:=Filtered(TriangulizedMat(eqs),x->not IsZero(x)); 2, AbelianInvariantsnewfree)AbelianInvariantsPreImagequotm)) mfpi:=GroupHomomorphismByImagesNC(newfree,m, (newfreeConcatenation, ListWithIdenticalEntries(dim,One(m)))); # first just try a bit, and see whether this gets all (e.g. if # module is irreducible). e:(mfpi,:heap)java.lang.StringIndexOutOfBoundsException: Index 70 out of bounds if e<>fail then step:=0; #extended presentation # Now write down the combined representation in wreath e:=java.lang.StringIndexOutOfBoundsException: Index 24 out of bounds for length # define map on subgroup. ifLength(1)0 Error"";; .primaryImages=mmutable List(GeneratorsOfGroup(newfree){[1..nn]}, x->ImagesRepresentative(e,x))); TrySecondaryImages(tab); i:=GroupHomomorphismByImagesNC(subr:=k-*[[i]]java.lang.StringIndexOutOfBoundsExc List(GeneratorsOfGroup(newfree){[1..nn]}, x-ImagesRepresentative,):; SetCosetTableFpHom(i,tab (,1-v2; e: # first just try a bit, and see whether this gets all (e.g. if e:=Intersection(e, KernelOfMultiplicativeGeneralMappingjava.lang.StringIndexOutOfBoundsException: I Assert(0,ForAll(RelatorsOfFpGroup(fps), ; GeneratorsOfGroup!quot)); if step=0 then i:=GroupHomomorphismByImagesNC(e!.quot,p, java.lang.StringIndexOutOfBoundsException: Index 36 out of bounds for length 10 =java.lang.StringIndexOutOfBoundsException: Index 47 out of bounds for length 47 j:=PreImage k:=List(eneratorsOfMonoid)java.lang.StringIndexOutOfBoundsException: if AbelianInvariants(j)=AbelianInvariants(newfree) (GeneratorsOfGroup( :function(is step: Info(InfoExtReps a:=*i; else Info(InfoExtRepsjava.lang.StringIndexOutOfBoundsException: Index 0 out of bounds e:=LargerQuotientBySubgroupAbelianization(mfpi,m: cheap:=false); e:Intersectione, x->ImagesRepresentative(e,x)):noassert fi; fijava.lang.StringIndexOutOfBoundsException: Index 21 out of bounds for length 2 step:=step+1; od; java.lang.StringIndexOutOfBoundsException: Range [8, 7) out of bounds for length else if simi= simi:=IsomorphismSimplifiedFpGroup(Sourcep=(w,; simiq:=InverseGeneralMapping(simi)*quot; fi; eargerQuotientBySubgroupAbelianization,) if e<>fail (); i:=simi*DefiningQuotientHomomorphism(e); j=((e)); j:=List(Orbits(j,MovedPoints(j)),x->Stabilizer(j,x[1])); j:=List(j,x->PreImage(i,x)); e:=Intersection(j); e (l,+1)fijava.lang.StringIndexOutOfBoundsException: Index 25 out of bounds for l ; fi e:(mfpim: if e<>fail then # can we do better degree -- greedy block reduction? nn:=e!.quot; if IsTransitive(nn,MovedPoints(nn)) then repeat ei:=ShallowCopy(RepresentativesMinimalBlocks(nn, MovedPoints(nn))); ,x->Length(); # long ones first j:=false; i:=1; whileofgens:FreeGeneratorsOfFpMonoid autom!.magemonwords=[];fijava.lang.StringInd str=(n, java.lang.StringIndexOutOfBoundsException: Index 24 out of bounds for len if Size(Core(nn,str))=1 then (fm,java.lang.StringIndexOutOfBoundsException: Index 54 out of bounds for length fi; i+java.lang.StringIndexOutOfBoundsException: Index 27 out of bounds for length 2 od; if j<>false then Info(InfoExtReps,4,"deg. improved by blocks ",Size(j)); # action on blocks i: imagemonwords]java.lang.StringIndexOutOfBoundsException: Index 31 out of bound j:=Size(nn); # make new group to not cache anything about old.,j,p,,; nn:=Group(List(GeneratorsOfGroup x->ImagesRepresentative(i,x)),()); SetSize(nn,j); fi; until j=false; fi; if not IsIdenticalObj(nn,e!.quot) then Info(InfoExtReps,2,"Degree improved by factor ", 1,; # power that changes the exponent sign :SubgroupOfWholeGroupByQuotientSubgroupjava.lang.StringIndexOutOfBoundsException Stabilizer(nn,1)); ; elif e=fail and IndexNC(p,m)>trysy then trysy=(;# never try again # can the Sylow subgroup get us something? m:=SylowSubgroup(p,prime); e:=PreImage inin [.mn Info(InfoExtReps,2,"Sylow testfor in =; p() only one TzGo :=(Rangei,,(m)1)java.lang.StringIndexOutOfBoundsException: Index 69 out of bound :i* j:=KernelOfMultiplicativeGeneralMapping( InverseGeneralMapping(e)*quot); i:=RandomSubgroupNotIncluding(Range(java.lang.StringIndexOutOfBoundsException: R Info(InfoExtReps,2,"Sylow found ",IndexNC(p,m)," * ", (Range)i); if # consider permdegree up to r,new; java.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0 # evaluate rules in images #e:=KernelOfMultiplicativeGeneralMapping( i java.lang.StringIndexOutOfBoundsExcept # DefiningQuotientHomomorphism(i));[,ReducedFormfrewnew); e:=fail; # not good fi; else Info(InfoExtReps,4,"Don't index ",Index( ord:java.lang.StringIndexOutOfBoundsEx fi; until<>; i:=p; quot:=DefiningQuotientHomomorphism( Concatenation(ListWithIdenticalEntries(mnfunc KernelOfMultiplicativeGeneralMapping(quot# for i in wrd do p:=Image(quot); Info(InfoExtReps,1,"index ",Index(i,m)," increases # Size(p)/Size(i)," at degree ",NrMovedPoints(p)); hom:=false; # we don't have hom cheaply any longer as group changed. # this is not an issue if module is irreducible it:=fail; simi:=fail; # cleanout info for first factor od; quot:=sim*quot; new:=GroupHomomorphismByImages(fp,p,GeneratorsOfGroup(fp), new)then List(GeneratorsOfGroup(fp),x->ImagesRepresentative(quot,fi fi; # if we used factor perm rep, be bolder if IsPermGroup(p) then new:=new*SmallerDegreePermutationRepresentation(p:cheap:=wasbold<>true); SetIsomorphismPermGroup(fp,new); elif IsPcGroup(p) new:=ReducedForm(frejava.lang.StringIndexOutOfBoundsException: R tIsomorphismPcGroup,ew; fi; ; if :(isc[i)2; SetIsSolvableGroup(fp,IsSolvableGroup(r.group)); fijava.lang.StringIndexOutOfBoundsException: Index 5 out of bounds for length 5 return fp; ); str:=List(ogens); local bas,ran,mats,o; if Length(coh.cohomology)=0 then return [coh.zero];fi; if Length(coh.cohomology)=1 and Size(coh.module.field gensmrep:List(eneratorsOfGroup # 1-dim space over GF(2) return [coh.zero,coh.cohomology[1]]; fi; # make basis #modulegenerators ran:=[BindGlobal("PermrepSemidirectModule"Gmodule) mats(p,-istoh.ohomology, v->SolutionMat(bas,coh.pairact(v in[.dim Orbits((ats)oh..^(coh.cohomologyjava.lang.StringIndexOutOfBoundsException: Index :Length(.); return o; end); ############################################################################# ## #M Extensions( G, M ) ## InstallOtherMethod(Extensionsod; true,IsGroup ,IsObject] -RankFilter( function(Gnew:=MappedWord(.presentation.[i,,ens1.]) local coh; (G,M; return(V(.eldoh.,oh.ero, x- r,gens ,(ctsBijective)java.lang.StringIndexOutOfBoundsException: Index 40 out o end); InstallOtherMethod(java.lang.StringIndexOutOfBoundsException: Index 35 out of bo true[ for j in.im do RankFilters ) function(G,M,P) ocal coh; coh:=TwoCohomologyGeneric(G,M); return fp:=frels; end);
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