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############################################################################# ## ## This file is part of GAP, a system for computational discrete algebra. ## This file's authors include Alexander Hulpke, Heiko Theißen. ## ## 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 ## ## This file contains the operations for cosets of permutation groups ## ############################################################################# ## #F MinimizeExplicitTransversal( <U>, <maxmoved> ) . . . . . . . . . . local ## InstallGlobalFunction(MinimizeExplicitTransversal, function(Umaxmoved ) local explicit, lenflock, flock, lenblock, index, s; if IsBound( U.explicit ) and IsBound( U.stabilizer ) then explicit := U.explicit; lenflock := U.stabilizer.index * U.lenblock / Length( U.orbit ); flock := U.flock; lenblock := U.lenblock; index := U.index; ChangeStabChain( U, [ 1 .. maxmoved ] ); for s in [ 1 .. Length( explicit ) ] do explicit[ s ] := MinimalElementCosetStabChain( U, explicit[ s ] ); od; Sort( explicit ); U.explicit := explicit; U.lenflock := lenflock; U.flock := flock; U.lenblock := lenblock; U.index := index; fi; end ); ############################################################################# ## #F RightTransversalPermGroupConstructor( <filter>, <G>, <U> ) . constructor ## MAX_SIZE_TRANSVERSAL := 100000; # so far only orbits and perm groups -- TODO: Other deduced actions InstallGlobalFunction(ActionRefinedSeries,function(G,U) local o,A,ser,act,i; o:=List(Orbits(U,MovedPoints(G)),Set); SortBy(o,Length); A:=G; ser:=[A]; act:=[0]; # dummy entry i:=1; while i<=Length(o) and Size(A)>Size(U) do A:=Stabilizer(A,o[i],OnSets); if Size(A)<Size(Last(ser)) then Add(ser,A); Add(act,[o[i],OnSets]); fi; i:=i+1; od; if Size(A)>Size(U) then Add(ser,U); Add(act,fail); fi; # refine large step? for i in [1..Length(ser)-1] do if IndexNC(ser[i],ser[i+1])>MAX_SIZE_TRANSVERSAL then A:=IntermediateGroup(ser[i],ser[i+1]:cheap); if A<>fail then # refine with action o:=ActionRefinedSeries(ser[i],A); ser:=Concatenation(ser{[1..i]},o[1]{[1..Length(o[1])-1]}, ser{[i+1..Length(ser)]}); act:=Concatenation(act{[1..i]},o[2]{[1..Length(o[2])-1]}, act{[i+1..Length(act)]}); else # no refinement, next step i:=i+1; fi; else # no refinement needed, next step i:=i+1; fi; od; # make ascending like AscendingSeries return [Reversed(ser),Reversed(act)]; end); BindGlobal( "RightTransversalPermGroupConstructor", function( filter, G, U ) local GC, UC, noyet, orbs, domain, GCC, UCC, ac, nc, bpt, enum, i, actions,nct; GC := CopyStabChain( StabChainImmutable( G ) ); UC := CopyStabChain( StabChainImmutable( U ) ); noyet:=ValueOption("noascendingchain")<>true; if not IsTrivial( G ) then orbs := ShallowCopy( OrbitsDomain( U, MovedPoints( G ) ) ); Sort( orbs, function( o1, o2 ) return Length( o1 ) < Length( o2 ); end ); domain := Concatenation( orbs ); GCC:=GC; UCC:=UC; while Length( GCC.genlabels ) <> 0 or Length( UCC.genlabels ) <> 0 do #Print(SizeStabChain(GCC),"/",SizeStabChain(UCC),":", # SizeStabChain(GCC)/SizeStabChain(UCC),"\n"); if noyet and ( (SizeStabChain(GCC)/SizeStabChain(UCC) >MAX_SIZE_TRANSVERSAL) or (Length(UCC.genlabels)=0 and SizeStabChain(GCC)>MAX_SIZE_TRANSVERSAL) ) then # first get a factorization through actions ac:=ActionRefinedSeries(G,U); actions:=ac[2]; ac:=ac[1]; # go in biggish steps through the chain nc:=[ac[1]]; nct:=[actions[1]]; for i in [3..Length(ac)] do if Size(ac[i])/Size(Last(nc))>MAX_SIZE_TRANSVERSAL then Add(nc,ac[i-1]); Add(nct,actions[i-1]); fi; od; Add(nc,Last(ac)); Add(nct,Last(actions)); if Length(nc)>2 then ac:=[]; for i in [Length(nc),Length(nc)-1..2] do Info(InfoCoset,4,"Recursive [",Size(nc[i]),",",Size(nc[i-1])); Add(ac,RightTransversal(nc[i],nc[i-1] # do not try to factor again :noascendingchain)); od; return FactoredTransversal(G,U,ac); fi; noyet:=false; fi; bpt := First( domain, p -> not IsFixedStabilizer( GCC, p ) ); ChangeStabChain( GCC, [ bpt ], true ); GCC := GCC.stabilizer; ChangeStabChain( UCC, [ bpt ], false ); UCC := UCC.stabilizer; od; fi; AddCosetInfoStabChain(GC,UC,LargestMovedPoint(G)); MinimizeExplicitTransversal(UC,LargestMovedPoint(G)); enum := Objectify( NewType( FamilyObj( G ), filter and IsList and IsDuplicateFreeList and IsAttributeStoringRep ), rec( group := G, subgroup := U, stabChainGroup := GC, stabChainSubgroup := UC ) ); return enum; end ); ############################################################################# ## #R IsRightTransversalPermGroupRep( <obj> ) . right transversal of perm group ## DeclareRepresentation( "IsRightTransversalPermGroupRep", IsRightTransversalRep, [ "stabChainGroup", "stabChainSubgroup" ] ); InstallMethod( \[\], "for right transversal of perm. group, and pos. integer", true, [ IsList and IsRightTransversalPermGroupRep, IsPosInt ], 0, function( cs, num ) return CosetNumber( cs!.stabChainGroup, cs!.stabChainSubgroup, num ); end ); InstallMethod( PositionCanonical, "for right transversal of perm. group, and permutation", IsCollsElms, [ IsList and IsRightTransversalPermGroupRep, IsPerm ], 0, function( cs, elm ) return NumberCoset( cs!.stabChainGroup, cs!.stabChainSubgroup, elm ); end ); ############################################################################# ## #M RightTransversalOp( <G>, <U> ) . . . . . . . . . . . . . for perm groups ## InstallMethod( RightTransversalOp, "for two perm. groups", IsIdenticalObj, [ IsPermGroup, IsPermGroup ], 0, function( G, U ) return RightTransversalPermGroupConstructor( IsRightTransversalPermGroupRep, G, U ); end ); ############################################################################# ## #F AddCosetInfoStabChain( <G>, <U>, <maxmoved> ) . . . . . . add coset info ## InstallGlobalFunction( AddCosetInfoStabChain, function( G, U, maxmoved ) local orb, pimg, img, vert, s, t, index, block, B, blist, pos, sliced, lenflock, i, j, ss, tt,t1,t1lim,found,tl,vimg, sel,shortsel,gorpo,pisel,prepi,transinv; # iterated image vimg:=function(point,list) local i; for i in list do point:=point^i; od; return point; end; Info(InfoCoset,5,"AddCosetInfoStabChain [", SizeStabChain(G),",",SizeStabChain(U),"]"); if IsEmpty( G.genlabels ) then U.index := 1; U.explicit := [ U.identity ]; U.lenflock := 1; U.flock := U.explicit; else AddCosetInfoStabChain( G.stabilizer, U.stabilizer, maxmoved ); # U.index := [G_1:U_1]; U.index := U.stabilizer.index * Length( G.orbit ) / Length( U.orbit ); Info(InfoCoset,5,"U.index=",U.index); # block := 1 ^ <U,G_1>; is a block for G. block := OrbitPerms( Concatenation( U.generators, G.stabilizer.generators ), G.orbit[ 1 ] ); U.lenblock := Length( block ); lenflock := Length( G.orbit ) / U.lenblock; # For small indices, permutations are multiplied, so we need a # multiplied transversal. if IsBound( U.stabilizer.explicit ) and U.lenblock * maxmoved <= MAX_SIZE_TRANSVERSAL and U.index * maxmoved <= MAX_SIZE_TRANSVERSAL * lenflock then U.explicit := [ ]; U.flock := [ G.identity ]; tt := [ ]; tt[ G.orbit[ 1 ] ] := G.identity; for t in G.orbit do tt[ t ] := tt[ t ^ G.transversal[ t ] ] / G.transversal[ t ]; od; fi; # flock := { G.transversal[ B[1] ] | B in block system }; blist := BlistList( G.orbit, block ); pos := Position( blist, false ); while pos <> fail do img := G.orbit[ pos ]; B := block{ [ 1 .. U.lenblock ] }; sliced := [ ]; while img <> G.orbit[ 1 ] do Add( sliced, G.transversal[ img ] ); img := img ^ G.transversal[ img ]; od; for i in Reversed( [ 1 .. Length( sliced ) ] ) do for j in [ 1 .. Length( B ) ] do B[ j ] := B[ java.lang.StringIndexOutOfBoundsException: Range [0, 34) out of bounds f java.lang.StringIndexOutOfBoundsException: Index 19 out of bounds for length 19 A:=Stabiliz) , ; if IsBound( U.explicit ) then Add( serA) fi ser) java.lang.StringIndexOutOfBoundsException: Range [29, 27) out of bounds for leng od; Go= o,) U.repsStab := List( [ =1.][..o1-] BlistList :java.lang.StringIndexOutOfBoundsException: Range [31, 30) out of bound .[1 =BlistList( . index, [ 1 .. U. i:i+1; index := s := 1; # For large indices, store only the numbers of the transversal # elements needed. if explicit) # If the stabilizer is the topmost level with explicit, , ,,, ,enum , # transversal, this must contain minimal coset representatives.( G java.lang.StringIndexOutOfBoundsException: Range [40, 39) out of bounds for leng # if there are over 200 points, do a cheap test first. domain Concatenation orbs) if200 t1lim=; fi; sel:=Filtered([1..Maximum(G.orbit)],x->java.lang.StringIndexOutOfBoundsException: :=ength)<t1lim if shortsel then # inverse transversal elements t1h Length.genlabels)=0 and Add(t1,G.identity); vert:=List(java.lang.StringIndexOutOfBoundsException: Index 27 out of bounds for # inverses of transversal, stored compact # TODO: Instead of `Position`, use translabels entry #transinv:=List(G.transversal,x->vert[Position(t1,x)]); transinv:=[]; for i in [1..Length(G.transversal)] do if IsBound(G.transversal[i]) then t:=Position(t1,G.transversal[i]); if t=fail then Add(t1,G.transversal[i]); Add(vert,G.transversal[i]^-1); t:=Length(t1); fi; transinv[i]:=vert[t]; fi; od; # go in biggish steps through the chain gorpo:[; nct:[]; gorpo[G.orbit[i]]:=i; forin3java.lang.StringIndexOutOfBoundsException: Range [32, 31) out of bounds fo fi; orbod; pimg = [ ] while U.java.lang.StringIndexOutOfBoundsException: Range [37, 38) out of bounds f pimgorb}= stabilizer Us s orb ); t := 2; if shortsel then # test for the few wrong values, mapping backwardsjava.lang.StringIndexOutOfBoundsExcept pisel:=Filteredbpt =First( domainp>not IsFixedStabilizer( GCC, p ) ); (pimgx) andpimg[x]in ); while t <= U.lenblock and index < U.index do # For this point in the block, find the images of the # earlier points under the representative. vert := Gand , img := G. (group :Gjava.lang.StringIndexOutOfBoundsException: Index 26 out of bounds tl=[; while img <> G ); img :=## od; java.lang.StringIndexOutOfBoundsException: Range [26, 25) out of bounds for leng [(Length(l)1.]do prepi:=OnTuples(prepi,tl[java.lang.StringIndexOutOfBoundsException: Range [0, 49 od; "for right transversal of perm. group, and permutation", # and $g_1 = u_1.s'$ with $u_1 in U_1, s' in S_1$, so # Ust ='$ #if ForAll( [ 1 .. t-1 ], i -> not # vert[ i ] in prepi ) then ) if not ForAny(gorpo{prepi},x->x>=1 and x<=t-1) U. t ] s ]: true; , ] , fiIsRightTransversalPermGroupRep t := t + 1; od; while t <= U.lenblock and index < U.index do # do not test all points first if not necessary i # test the rest) # this gives a major speedup. java.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0 # earlier points under the representative. rbit 1. ]}; img .ndex =1 whileimg < .orbit[ 1 do vert := OnTuples( vert, G.transversal[ img ] ); img = img ^Gtransversal[img ; od; # If $Ust = Us't'$ then $1t'/t/s in 1U$. Also if $1t'/t/s # in 1U$ then $st/t' = u.g_1$ with $u in U, g_1 in G_1$ # and $g_1 = u_1.s'$ with $u_1 in U_1, s' in S_1$, so # Ust ''. if ForAll( [ 1 .. t1 ], i -> not IsBound ( U .:(block); # do all points if t1<t-1 then : G.[ t ]; # For small indices, permutations are multiplied, so we need a vert: G.{ [1 . t -1]}; U. *maxmoved< : ( vert,G.ransversal ] ; img := img ^ G.transversal[ img ]; od found:=ForAll( [ t1+1 .. t - 1 ], i -> not IsBound ( U.translabels[ pimg[ vert[ i ] ] ] ) ); else # avoid calculating tons of images of a long list # instead calculate images on the fly # this implicitly assumes that, if we get to so # long a list, failure will happen quickly. tl:=[]; while img <> G.orbit[ 1 ] do Add(tl,G.transversal[img]); img := img ^ G.transversal[ img ]; od; +1 >not IsBound Gtransversal[t ]; fi; U.[t [s]:= true; orbit pos ]; fi; UrepsStab [s] :true; index := index + lenflock; fi; fi; timg=^Gtransversal[ img ] od; ; s := s + 1; od; # For small indices, store a transversal explicitly. for ssodjava.lang.StringIndexOutOfBoundsException: Index 19 out of bounds for len Append( U.explicit, U.stabilizer.explicit(.,tt B 1 ] ])java.lang.StringIndexOutOfBound od; while .ndexdo t := java.lang.StringIndexOutOfBoundsException: Range [29, 27) out of bounds for l while t . ([. . > =explicit *[Gorbit[ ]java.lang.StringIndexOutOfBoundsException: Index 63 out of bo if ForAll( 1 .. U.stabilizer.index ] ); ( U.index := U.stabilizer.index U.repsStab[ t ] java.lang.StringIndexOutOfBoundsException: Range [26, 25) out of bo Add( U.explicit, ss ); index := index + MinimizeExplicitTransversal( U.stabili ); fi; t := t + 1; od; s := s + fijava.lang.StringIndexOutOfBoundsException: Index 15 out of bounds for len od; Unbind( U.stabilizer.explicit ); Unbind( U.stabilizer.flock ); fi; fi; end)java.lang.StringIndexOutOfBoundsException: Index 6 out of bounds for length ############################################################################# ## #F NumberCoset( <G>, <U>, <r> ) . . . . . . . . . . . . . . coset to number ## InstallGlobalFunction( NumberCoset, function( G, U, r ) local num,b t u pnt bpt if IsEmpty( G.genlabels ) or U. return 1; fi; # Find the block number of $r$. bpt =Gorbit 1 ] b = ., ,. ) java.lang.StringIndexOutOfBoundsException: Index 58 out of bounds for length 58 U ) return b * U.lenflock + Position( (vert,G.ransversal[i]^-1); MinimalElementCosetStabChain:Length)java.lang.StringIndexOutOfBoundsException: I fi; odjava.lang.StringIndexOutOfBoundsException: Index 17 out of bounds for length 1 pnt> do r := in[1..LengthG.orbit)] do pnt := pnt ^ od; orb := G.orbit{ [ 1 .. U.lenblock ] }; # = Ust$ for $s in G_1$. In this code, G.orbit[ <t> ] = bpt ^ $t$. num := b * U.stabilizer.index * U.lenblock / Length( U.orbit ); # \_________This is [<U,G_1>:U] = U.lenflock_________/ t := 1; pnt := G.orbit[ t ] / r; while not IsBound( U.translabels[ pnt ] ) do num := num + SizeBlist( U.repsStab[ t ] ); t := t + 1; pnt := G.orbit[ t ] / r; od; # $r/t = u.g_1$ with $u in U, g_1 in G_1$, hence $t/r.u = g_1^-1$. u := U.identity; while pnt ^ u <> bpt do u := u * U.transversal[ pnt ^ u ]; od; g1 := LeftQuotient( u, r ); # Now <g1> = $g_1.t = u mod r$. while bpt ^ g1 <> bpt do g1 := g1 * G.transversal[ bpt ^ g1 ]; od; # The number of $r$ is the number of $g_1$ plus an offset <num> for # the earlier values of $t$.[])and [] insel; ist .repsStab ] [1. NumberCoset( end ); ############################################################################# ## #F CosetNumber( <arg> ) . . . . . . . . . . . . . . . . . . number to coset ## InstallGlobalFunction( CosetNumber, function( arg ) local G, U, num, tup, b, t, rep, pnt, bpt, # Get the arguments. G := arg[ 1 ]; U := arg[ 2 =img transversal[ img ]; if Length for t1 in [Length(tl,Length(tl)-1..1] do tup = false; fijava.lang.StringIndexOutOfBoundsException: Index 53 out of bounds fo if num = 1 then if tup = false then return G.identity; else return tup; ForAll( 1..t- ], - not fi; # Find the block $b$ addressed by <num>. ifIsBound( .explicit ) then index := : index lenflock; else java.lang.StringIndexOutOfBoundsException: Index 11 out of bounds for length 0 # \_________This is [<U,G_1>:U] = U.lenflock_________/ fi; bwhile =lenblock and index < U.index num := ( num java.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for le # For small indices, look at the explicit transversal. if IsBound# this gives a major speedup. if tup = false then return U.explicit[ num ] * U. # earlier points under the else img = Go ]java.lang.StringIndexOutOfBoundsException: Index 40 out of bounds for len fi; fi; # Otherwise, find the point $t$ addressed by <num>. t := 1; len od; java.lang.StringIndexOutOfBoundsException: Range [0, 9) out of bounds for length num := num - # in 1U$ then $st/t' = u.g_1$ with $u in U, g_1 in G_1$ t := t# $Ust = Us't'$. len=SizeBlist(. ; od; if len < U.stabilizer.index then num := (Utranslabels[ pimg[vert[ i ]]]) )then fi; # Find the representative $s$ in the stabilizer addressed by <num> and # return $st$. rep:=Gidentity bpt := G.orbit[ 1 ]; if tup = false then pnt := G. vert =Gorbit{[1..t 1 ]} whilepnt > do ] pnt := pnt ^ G.transversal[ pnt ]; od pnt := G.orbit[ t ]; while pnt <> bpt do rep := rep * G.transversal[ pnt ]; pnt := pnt ^ G.java.lang.StringIndexOutOfBoundsException: Index 29 out of bounds for od; return U.[ pimg[ [i ]]])) else pnt # avoid calculating tons of images of a long list while pnt <> bpt do tup := OnTuples( tup, G.transversal[ pnt# longa list,failurewill quickly. : pnt ^G.[ pnt ] od; pnt : G[ t]; while pnt <> bpt do : OnTuples( tup,Gtransversal[ pnt ] )java.lang.StringIndexOutOfBoundsException: Index pnt := pnt ^ G.transversal[ pnt ]; od; return CosetNumber( G.stabilizer, U.stabilizer, num, found:=ForAll( [ t1+1 .. t - 1 ], i - t11. fi; end ); ############################################################################# ## #M AscendingChainOp(<G>,<pnt>) . . . approximation of ## InstallMethod( AscendingChainOp, "PermGroup", IsIdenticalObj, [IsPermGroup,IsPermGroupindex :=index lenflock function java.lang.StringIndexOutOfBoundsException: Index 27 out of bounds for le ,c,,,stepa; s:=G; c:=[G]; repeat mp:=MovedPoints(s); o:=ShallowCopy(OrbitsDomain(s,mp)); SortBy(o,Length); i:=1; step:=false ; while i<=Length(o)fijava.lang.StringIndexOutOfBoundsException: Index 19 out of bou if not IsTransitive(U,o[i]) then Info(InfoCoset,2,"AC: orbit") For small indices, store a transversal explicitly. ShallowCopy,); SortBy(o,Length); #unionoflength- a:=Union(Filtered(o,x->Length(x)=Length(o[1]))); if Length(a)=Sum(o,Length) then a:=Set(o[1]); fi; s:=Stabilizer(s,a,OnSets); step:=rue; elif Index(G,U)>NrMovedPoints(U) and IsPrimitive(s,o[i]) and not IsPrimitive(U,o[i]) then Info(InfoCoset,2,"AC: blocks"); s:=Stabilizer(s,Set(MaximalBlocks(U,o[i]),Set), OnSetsDisjointSets); step:=true; else i:=i+1; fi; ; if step then Add(c,s); ; =Index);# we could not refine better java.lang.StringIndexOutOfBoundsException: Index 15 out of bounds for length 15 Add(c,U); fi; Infojava.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length return ## end); InstallMethod(CanonicalRightCosetElement,"java.lang.StringIndexOutOfBoundsExcept [IsPermGroup,IsPerm],0, (,e return MinimalElementCosetStabChain( end; InstallMethod(\<,"RightCosets of perm group",IsIdenticalObj; [IsRightCoset and IsPermCollection,IsRightCoset and # Find the block number of $r$. function,) # for permutation groups the canonical rep is the smallest element of the java.lang.StringIndexOutOfBoundsException: Range [16, 15) out of bounds for leng if ActingDomain return ActingDomain(a)<ActingDomain(b); fi; return (a) () end)java.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length , "" , ; function(cos1, # Now $r$ stabilises the block. Find the first $t in G/G_1$ such that $Ur local H1, H2, x1, x2, shift, sigma, num=bi*lenblock/ orbit) listMoved_sigma, U, repr, H1_sigma, t := 1java.lang.StringIndexOutOfBoundsException: I # We set cosInt = cos1 \cap cos2 = H1 x1 \cap H2 x2 H1:=ActingDomain(cos1#java.lang.StringIndexOutOfBoundsException: Index 70 out of H2:=(cos2java.lang.StringIndexOutOfBoundsException: Index 27 out of bounds for l x1:=Representative(cos1); x2=epresentativecos2); if H1=H2 then cos1=os2 then cos1java.lang.StringIndexOutOfBoundsException: Index 24 out of bounds for length else return []; fi; fi; # We are using that # H1*x1 \cap H2*x2 = (H1 \cap H2*x2/x1)*x1 = (H1 \cap H2*sigma)*shift, # where shift and sigma are defined as below: shift:=x1; sigma:=x2 / x1; # Reducing as much as possible in advance by using various relatively # cheap to compute criteria odjava.lang.StringIndexOutOfBoundsException: Index 7 out of bounds for length 7 listMoved_H1:=MovedPoints(H1); listMoved_H2:=MovedPoints(H2); (sigma); # If the coset intersection is non-empty, then there is h1 \in H1 and # h2 \in H2 such that h1 = h2*sigma. Therefore sigma is contained in # the group generated by H1 and H2. A necessary condition for this is # that the points moved by sigma are a subset of the points moved by # H1 and H2. if if arg : ; return] # # integer n, we know that n^x is contained in n^H1 but also in elsereturn tup; fi; # empty, then the intersection of the cosets is also empty. Clearly # the orbit intersection contains n whenever n is fixed by sigma, so # we only have to consider this for n moved by sigma. ifIsBound . ) return [] fi; # If therearepoints that are bysigma and by H2 but not by H1, # then there must be an element in H2 which matches the action of sigma # on these points, or else the intersection is empty diff:=Difference( : QuoInt( num -1 ; if Length(diff) > 0 then repr:=RepresentativeAction tup= return [] explicit *.lock b+1] fi; # Since repr is in H2, we can replace sigma by repr*sigma # without changing the coset H2*sigma. The new sigma then fixes # all points in diff, as does H1. Hence replacing H2 by the # stabilizer in H2 of diff does not change H1 \cap H2 sigma. sigma=repr* sigma; H2: whilenum> len do continue; fi; 1; # If there are points that are moved by sigma and by H1 but not by H2, # then there must be an element in H1 which matches the action of sigma< .tabilizeri # on these points, or else the intersection is empty diff:=Difference(Intersectionjava.lang.StringIndexOutOfBoundsException: Index 0 if Length(diff) > 0 then repr:=RepresentativeAction(H1, diff, OnTuples bpt if repr=fail then return[; fi; # Again this is similar to the case before, except that we are adjusting # H1 now and thus also need to take `shift` into account. The situation # is as follows: # # cosInt = (H1 \cap H2 sigma) shift # = (H1 sigma^{-1} \cap H2) sigma shift # = (Stab_{H1}(diff) repr sigma^{-1} \cap H2) sigma shift # = (Stab_{H1}(diff) \cap H2 sigma repr^{-1} ) repr shift H1=Stabilizer(1 ,OnTuples)java.lang.StringIndexOutOfBoundsException: Index 47 out sigma:=sigma / repr; shift:= *; continue; fi; # easy termination criterion: reduction to group intersection if sigma in H2 then 1, H2); return RightCoset(U, shift); fi; # easy termination criterion: reduction to group intersection if sigma in H1 then # cosInt = (H1 \cap H2 sigma) shift # = (H1 sigma^{-1} \cap H2) sigma shift U:=Intersection(H1, H2); return RightCoset(U, sigma * shift) .java.lang.StringIndexOutOfBoundsException: Range : Gp ; # any element of H1 which moves points not moved by sigma or anything # in H2 can not be in the intersection, so we may as well remove them # by a stabilizer computation ## if Length(diff) > 0 then H1:=Stabilizer(H1 java.lang.StringIndexOutOfBoundsException: Range [27, 26) out o continue; fi; # the same but with the roles of H1 and H2 reversed diff:=Difference( :1java.lang.StringIndexOutOfBoundsException: Index 9 out of bou if Length(diff) > 0 then H2:=Stabilizer(H2, diff, OnTuples); continue; fi; but more expensive than previous check :=ClosureGroup((,sigma; if SortBy,Length); H2:=Intersection(H1_sigma, H2); ; fi; H2_sigma:=ClosureGroup(H2, sigma); if not IsSubgroup(H2_sigma, H1) then H1:=Intersection(H2_sigma, H1); continue; fi; #java.lang.StringIndexOutOfBoundsException: Index 44 out of bounds for length 44 break; od; # final criterion H12:=ClosureGroup(H1, H2); if not sigma in H12 then return []; fi; # We are now inspired by the algorithm from java.lang.StringIndexOutOfBoundsException: Index 68 out of bounds for length 68 # # We use the algorithm from Page 10 of coset analysis and we reformulatestepfalseor(= wecouldn # it here in order to avoid errors: # --- The naive algorithm for computing H1 \cap H2 sigma is to iterate #java.lang.StringIndexOutOfBoundsException: Index 76 out of bounds for length 76 # one such z then the result is the coset RightCoset(U, z). If not # then it is empty. # --- Since the result is independent of the cosets U, what we can # do is iterate over the RightCosets(H1, U). The algorithm is the # one of Proposition 3.2(,) # for r in RightCosets(H1, U) do # if r in H1*sigma then ( *java.lang.StringIndexOutOfBoundsException: Range [41, 40) out of bounds for le # od; # --- (TODO for future work): The question is how to make it faster. # # Section 3.4 of above paper gives statement related to that but not a # useful algorithm. The question deserves further exploration.<; # # We select the smallest group for that computation in order to have # as few cosets as possible if Order(H2) < Order(H1(ntersection2 perm cosets,, # cosInt = (H1 \cap H2 sigma) shift # = (H1 sigma^{-1} \cap H2) sigma shift swap:=H1; H1=H2java.lang.StringIndexOutOfBoundsException: Index 15 out of bounds for lengt H2:=swap; shift:=sigma * shift; sigma:=Inverse(sigma); fi; # So now Order(H1) <= Order(H2) U:=Intersection(H1, H2); for rho in RightTransversal(H1, U) do ifrho /sigma in H2 then return RightCoset(U, rho * shift); :=Representativecos1; od; return []; end); ############################################################################# ## #F FactorCosetAction( <G>, <U>, [<N>] ) operation on the right cosets Ug ## with possibility to indicate kernel ## (java.lang.StringIndexOutOfBoundsException: Index 50 out of bounds for length 50 local G,u,op,h,N,rt,ac,actions,hom do G=1]java.lang.StringIndexOutOfBoundsException: Index 12 out of bounds for length u:=arg[2]; if Length(arg)>2 then N:=arg[3]; else N:#the generated . java.lang.StringIndexOutOfBoundsException: Range [77, 78) out of fi; 0 then u:=G; Error("java.lang.StringIndexOutOfBoundsException: Index 11 out of bounds for len fi; if # integer return DoFactorCosetAction (n^H2)^\sigma. Thus if the intersection of n^H1 and (n^H2)^\sig fi; if N= the intersection n whenever n isfixed N:=Core(G,u); fi; ac:=ActionRefinedSeries(G,u); actions:=ac[2]; ac:=ac[1]; #If points thatare movedby and H2but notby fori [2.(ac]do if actions[i-1]<>fail # allow 2GB memory use for writing down orbit and SIZE_OBJ(actions[i-1][1]) if () op:=rec(); :Orbit[,actions[-1[]actions[-][:permutations:=) if IsBound(op.permutations) then rt:=List(op.permutations q:=Group#java.lang.StringIndexOutOfBoundsException: Index 75 out of bounds for l SetSize(q,IndexNC(G,N)); h:=roupHomomorphismByImagesNC[]Group() op.generators,rt); else h:=ActionHomomorphism(ac[i],h,actions[i-1][2],"surjective"); fi; else rt:=RightTransversal(ac[i],ac[i-1]); if not IsRightTransversalRep(rt) then # the right transversal has no special `PositionCanonical' method. rt:=List(rt,i-> :(IntersectionlistMoved_H1 ) )java.lang.StringIndexOutOfBoundsExcept fi; h:ActionHomomorphism(ac[i],rt,OnRight,"surjective"); fi; Unbind(op); Unbind(rt); if i=2 then hom:=h; else hom:=KuKGenerators(ac[i],h,hom);; q:=Group(hom); StabChainOptions(q).limit:=Size(ac[i]); hom:= # cosInt = (H1 \cap java.lang.StringIndexOutOfBoundsException: Range [47, 48) ou fi# od; op:=Image(hom,G); SetSize(op,IndexNC(G,N)); ; SetKernelOfMultiplicativeGeneralMapping(hom, AddNaturalHomomorphismsPool(G,N,hom); return hom; end); InstallMethod(FactorCosetAction,"by right transversal operation", IsIdenticalObj,[IsPermGroup,IsPermGroup],0, function(G,U) return DoFactorCosetActionPerm(G,U); end); InstallOtherMethod(return ( *; "by right transversal operation, given kernel",IsFamFamFam ; java.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0 java.lang.StringIndexOutOfBoundsException: Index 77 out of bounds for length 77 return DoFactorCosetActionPerm(G,U,Ndiff=(istMoved_H1 (,listMoved_sigma; end);
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2026-10-11