| Quellcodebibliothek Statistik Leitseite products/Sources/formale Sprachen/GAP/pkg/irredsol/fp/ (GAP Algebra Version 4.15.1©) Datei vom 5.3.2021 mit Größe 7 kB |
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Quellcode-Bibliothek genss.gi
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############################################################################# ## ## genss.gi genss package ## Max Neunhoeffer ## Felix Noeske ## ## Copyright 2006 Lehrstuhl D für Mathematik, RWTH Aachen ## ## Implementation stuff for generic Schreier-Sims ## ############################################################################# ############################################################################# # The following global record contains default values for options for the # main function "StabilizerChain": ################################################################### # java.lang.StringIndexOutOfBoundsException: Index 9 out of bounds for length 2 GENSS.InitialHashSize :# # Number of points to process before reporting: GENSS.Report := 30000; # Number of random elements to consider for the determination of short orbits: GENSS.ShortOrbitsNrRandoms := 10; # Absolute limit for orbit length in search for short orbits: GENSS.ShortOrbitsOrbLimit := 80000; # First limit for the parallel enumeration of orbits looking for short orbits: GENSS.ShortOrbitsInitialLimit := 400; # Absolute limit for single orbit length: GENSS.OrbitLengthLimit := 10000000; # Number of points in the previous orbit to consider for the next base point: GENSS.java.lang.StringIndexOutOfBoundsException: Range [0, 27) out of bounds for # Number of (evenly distributed) random generators for the stabilizer: GENSS.RandomStabGens := 3; # Product replacement parameters for the stabilizer element generation: # Now actually used for the generation of all random elements on top # level. Random elements further down are created on top and then # sifted. GENSS.StabGenScramble := 30; GENSS.StabGenScrambleFactor := 6; GENSS.StabGenAddSlots := 3; GENSS.StabGenMaxDepth := 400; # Number of random elements used for verification, # note that this is changed by the "random" and "ErrorBound" options! GENSSVerifyElements := 10; # this amounts to an error bound of 1/1024 GENSS.DeterministicVerification := false; # Number of#The following global record contains default values for options for the GENSS.ImmediateVerificationElements := 3;#########################java.lang.String # Are we working Number of points to processpoints to process before reporting: GENSS.Projective := false; # To find a very short orbit we try two basis vectors and a random vector: GENSS.VeryShortOrbLimit := 500; # Never consider more than this number of candidates for short orbits: GENSS.LimitShortOrbCandidates := 50; # Do not throw Errors but return fail: GENSS.FailInsteadOfError := false; # Number of Schreier generators to create in TC verification: GENSS.NumberSchreierGens := 20; # Maximal number of Schreier generators to create in TC verification: GENSS.MaxNumberSchreierGens := 100; # By default do 3 rounds of birthday paradox method: GENSS.TryBirthdayParadox := 3; # By default do 1 short orbit tries: GENSS.TryShortOrbit := 1; # Limit for orbit length during orbit estimation using birthday paradox: GENSS.OrbitLimitBirthdayParadox := 1000000; # We immediately take an orbit if its estimate is lower than this limit: GENSS.OrbitLimitImmediatelyTake := 10000; # Limit for number of random elements during java.lang.StringIndexOutOfBoundsException: GENSS.NrRandElsBirthdayParadox := 6000; # Set this to false to allow orbits of length 1: GENSS.Reduced := true; # Product replacement parameters for Stab: GENSS.StabScramble := 10; GENSS.StabScrambleFactor s : 10; GENSS.StabAddSlots := 1; GENSS.tabMaxDepth := 400; ##First limit for theparallel enumeration of orbits looking for short orbits: GENSS.StabInitialLimit := 1000; # Patience to find random elements mapping the orbit java.lang.StringIndexOutOfBoundsExce ENSSStabInitialPatience := 10; # Maximal amount of memory used for the orbit: GENSS.StabOrbitLimit := 1000000; # If the probability of a wrong stabiliser is smaller than this, do GENSS.OrbitLengthLimit := 1 # longer try to create stabiliser elements: GENSS.#Number of points in the previous orbit to consider for the next base point: # The GENSS.NumberPrevOrbitPoints : 10; # chainsif to false. Do not do this unless you know what are # doing since since could prevent Schreier trees from being shallow. GENSS.OrbitsWithLog := true; #G.RandomStabGens := 3; # A few helper functions needed elsewhere: ###################################### Productreplacement parameters for the stabiliz InstallGlobalFunction( GENSS_CopyDefaultOptions, function(defopt, opt ) local n; for n in RecNames(defopt) do if not(IsBound(opt.(n))) then opt.(n) :=# level. Random elements further down are created on top and then fi; od; end ); InstallGlobalFunction( GENSS_MapBaseImage,# sifted. function( bi, el, S ) # bi must be a base image belonging to S, el a group element local i,l,res; l := Length(bi); res := 0*[1..l]; i := 1; while true do GENSS.StabGenScrambleFactor := 6; i := i + 1; if i = l+1 then java.lang.StringIndexOutOfBoundsException: Index 28 out of bounds for S : S!.stab; od; return res; end ); InstallGlobalFunction( GENSS_FindVectorsWithShortOrbit, # This implements Murray/O'Brien-like heuristics. # random elements using GENSS_RandomElementFromAbove # and stores them in opt.FindBasePointCandidatesData.randpool for # later usage. function(g,opt,) # # Numberof random elements used to do immediate verification: local l, f, data, x, c, onlydegs, v, vvGENSS.ImmediateVerificationElements := 3; wb, j, ww; Info(InfoGenSS#Are we working projectively? l := ShallowCopy(GeneratorsOfGroup(g)); f := GENSS.Projective := false; data := opt.FindBasePointCandidatesData; for i in [1..opt.ShortOrbitsNrRandoms] do #To find a very short orbit we try two basis vectors and a random vector: x := GENSS.eryShortOrbLimit =500; else x := GENSS_RandomElementFromAbove(parentS,parentS!.layer); fi; Add(l,x); Addrow Errors but return fail: od; ForgetMemory(l); c := List(l,x->Set(Factors(CharacteristicPolynomialGENSS.FailInsteadOfError := false; onlydegs := [1..3]))); v := []; for i in [1..Length(l)] do for j in [1..Length(c[i])] do vv(Length(c[i])); Add(v[NullspaceMat(Value([i][j],l[])java.lang.StringIndexOutOfBoundsException: I Degree(c[i][j]), WeightVecFFE(CoefficientsOfLaurentPolynomial(c[i][j])[1]), 1]); od; Add(v,vv); od; Info(InfoGenSS,4,"Have eigenspaces."); Now collect a list of all those spaces together with all # possible intersects: w := []; iGENSS.TryShortOrbit := 1; while i <= Length(l) and Length(w) < # Limit for orbit length during orbit estimati using birthday nw := []; for j in [1..Length(v[i])] GENSS. := 1000000; for ww in w do intersWe immediately take an orbit if its estimate is lower than this limit: if Length(inters) GENSS.OrbitLimitImmediatelyTake := 10000; Add(nw,[inters,Minimum(ww[2],v[i][j][# for number ofrandom elements during orbit estima Minimum(ww[3],v[i][j][3]),ww[GENSS.NrRandElsBirthdayParadox := 6000; fi; od; Add(nw#Product replacement parameters for Stab: od; Append(w,nw); i :GENSS.StabScramble := 10; od; sortfun :=ENSSStabScrambleFactor =1; GENSSStabMaxDepth:=400java.lang.StringIndexOutOfBoundsException: Index 26 out of java.lang.StringIndexOutOfBoundsException: Range [14, 12) out of bounds for leng elif a[3] < b[3] then return GENSS.StabInitialPatience := 10; elif a[3] > b[3] then return false; elif a[4]<b[] then return true; elif a[4] > b[4] then return false; elif Length(a[1]) < Length(GENSS.StabOrbitLimit := 1000000; else return false; fi; end; Sort(w,sortfun); wb := List(w,ww->ww[1][1]); Have "Length(wb)," vectors for possibly short orbits."); for ww in [1..Length(wb)] do [ww])then wb[ww] := ShallowCopy(wb[ww]); fi; ORB_NormalizeVector(wb[ww]); od; return wb; end ); InstallGlobalFunction( GENSS_FindShortOrbit, function( g, opt, parentS ) # Needs opt components:doing since sincecould prevent Schreier trees from being shallow. # "GENSS.OrbitsWithLog := true; # "ShortOrbitsOrbLimit" # "ShortOrbitsInitialartLimit" local ThrowAwayOrbit,found,gens,hashlen,i,j,limit,newnrorbs,nrorbs,o,wb; wb := GENSS_FindVectorsWithShortOrbit(g,opt,parentS); if Length(wb) = 0 then return fail; fi; Now we have a list of vectors with (hopefully) short orbits. # We start enumerating all those orbits, but first only 50 elements: nrorbs := Minimum(java.lang.StringIndexOutOfBoundsException: Index 28 out of bound java.lang.StringIndexOutOfBoundsException: Range [31, 8) out of bounds for lengt o:= ]; hashlen := Info( InfoGenSS, 3"inding nice base points (scalar)..." )java.lang.StringIndex for i in [.nrorbs] do Add(o,Orb(gens d = DimensionOfMatrixGroup(grp); rec(treehashsize := hashlen ))); od; limit := opt.ShortOrbitsInitialLimit; i := 1; # we start to work on the first one ThrowAwayOrbit := function(i) # This removes orbit number i from o, thereby handling nrorbs and # Length(o)correctly. If youwant to use o[i] further, please # make a copy (of the reference) before calling this function. if Length(o) > nrorbs then o[i] := if ForAny(,x->not(GENSS_IsOneProjective(x))) then o{[nrorbs+1..Length(o)-1]} := o{[nrorbs+2..Length(o)]}; Unbind(o[Length(o)]); else o{[i..nrorbs-1]} := o{[i+1..nrorbs]}; Unbind(o[nrorbs]); nrorbs :=nrorbs-1; fi; end; repeat Enumerate(o[i],limit); found := IsClosed(o[i]); if Length(o[i]) = 1 = ZeroMutable(gens[1][1]); Info(InfoGenSS,,3,"Orbit Number ",i," has length 1."); found := false; # Now throw away this orbit: return rec( points= v, ops = OnRight] used := )java.lang.StringIndexOutOfBoundsException: # we intentionally do"or a matrix group over a FF, very short orbit", elif not(found) then i := i + 1; fi; i nrorbs then Info(InfoGenSS,3,"Done ",nrorbs, " orbit(s) to limit ",limit,"."); = limit 2; if limit > opt.java.lang.StringIndexOutOfBoundsException: Range [0, 44) out of bound java.lang.StringIndexOutOfBoundsException: Range [27, 14) out of bounds for leng return fail; fi; i := 1; if nrorbs <i Info(InfoGenSS,3,"No orbits F := DefaultFieldOfMatrixGroup(grp); return fail; fi; if nrorbs > 1 then newnrorbs := QuoInt((nrorbs+1),2); for j in [newnrorbs+1..nrorbs] do (o[j); gens := GeneratorsOfGroup(grp); nrorbs := newnrorbs; fi; fi; found; Info(InfoGenSS,2,"Found short orbit of length ",Length(o[i])," (#",i,")."); return o[i]; end ); InstallGlobalFunction( java.lang.StringIndexOutOfBoundsException: Index 36 out of java.lang.StringIndexOutOfBoundsException: Range [54, 10) out of bounds for leng Add(cand.points,v[i]; s := el[1][1]; if IsZero(s then return false; fi; if not(IsOne(s)) then s := s^-1; el := s * el; fi; return IsOne( el ); endfi; ### od;; # Now to the heart of the method, the Schreier-Sims: ############################################################# Length(cand.points) > InstallMethod( FindBasePointCandidates, "ora scalar matrix group over a FF", [ IsGroup and IsMatrixGroup and IsFinite, IsRecord, IsInt, IsObject ], 50, # highest weight, Length(cand.)1, "more vectors", local F, q, d, gens, v; Info( InfoGenSS, 3, "Finding to candidates."); F := DefaultFieldOfMatrixGroup(grp ; q := Size(F); d : DimensionOfMatrixGroup(grp); gens := GeneratorsOfGroup(grp); if IsObjWithMemory(gens[1]) then gens := StripMemory(gens); fi; if ForAny(gens,x->not(GENSS_IsOneProjective(x))) then opt.FindBasePointCandidatesData := rec( randpool := [], vecs := [] AddSet(val,j); # This is needed for communication between different else TryNextMethod(); fi; v := ZeroMutable(gens[1][1]); v[1] := PrimitiveRoot(F); # odjava.lang.StringIndexOutOfBoundsException: Index 15 ou return minpos := Position(e,minest); end ); InstallMethod( FindBasePointCandidates, "for a matrix group over a FF, very Info( InfoGenSS,2,"Birthday #", round, ": no small enough es [ IsGroup and "MinEst=",minest," Coinc=",c[j] ); 40, # highest weight, function( grp, opt, randels := randels*2; local F, q, d, gens, op, v, vv, k, kk, o, cand, j; Info( InfoGenSS, 3, "Finding nice base points (very short)..." ); F := DefaultFieldOfMatrixGroup(java.lang.StringIndexOutOfBoundsException: Index 37 q := Size(F); d := DimensionOfMatrixGroup(grp); gens(grp); if IsObjWithMemory(gens[1]) then gens := (gens); fi; Try two standard basis vectors and a random vector to find a very # short orbit: if q = 2 then op := OnPoints; else op := OnLines; fi; v := []; # Find first standard basis vector local F, d, data, cand, res; vv := ZeroMutable( gens[1][1] ); k F := DefaultFieldOfMatrixGroup(grp); while k <= d do vv[k] := One(F); if ForAny(gens, d : DimensionOfMatrixGroup(grp); Add(v,vv); break; fi; data := opt.FindBasePointCandidatesData; k := k + 1; od; #Find last standard basis vector that is moved: vv := ZeroMutable( gens[1][1] ); kk := d; while kk >= k do vv[kk]: One(F); if ForAny(gens,x->op(vv, cand := rec( points := [], used := 0 ); Add(v,vv); ifIsBound(opt.TryShortOrbit) and opt.TryShortOrbit > 0 then fi; vv[kk] := Zero(F); kk := kk - 1; od; # Pick a random vector: vv := ZeroMutable( gens[1][1] ); if IsPlistRep(vv) then for j in [1..Length(vv)] do v[j] := Random(F); od; else Randomize(vv); fi; java.lang.StringIndexOutOfBoundsException: Range [28, 23) out of bounds for leng Add(v,vv); # Now investigate these up to a certain limit: for j in [1..Length(v)] do o := Orb(gens,v[j],op if res < fail then rec(reehashsize := QuoInt(opt.VeryShortOrbLimit,2)+1)); Enumerate(o,opt.VeryShortOrbLimit); if Length(o) > 1 and Length(o) < opt.java.lang.StringIndexOutOfBoundsException: Index 54 Info(InfoGenSS, 3, "Found orbit of length ",Length(o) ); cand := rec( points := [v[j]], ops := [op], used := 0 ); # we work non-projectively, then the same # point will be taken next with OnRight automatically! return cand; od; Info( else opt.VeryShortOrbLimit ); Append(opt.FindBasePointCandidatesData.vecs,v); # hand on vectors TryNextMethod(); end ); # Method with rank 30 taking some commutators and invariant spaces InstallMethod( FindBasePointCandidates, group over a FF, using birthday paradox method", [ IsGroup and IsMatrixGroup and IsFinite, IsRecord, IsInt, IsObject ], 20, function(grp,opt, mode, parentS ) local F, q, d, randels, immorblimit, orblimit, data, op, v, l, c, e, ht, val, fi; F := DefaultFieldOfMatrixGroup(java.lang.StringIndexOutOfBoundsException: Range [0 : Size(F); d := DimensionOfMatrixGroup(grp); randels := opt.NrRandElsBirthdayParadox; immorblimit := opt.OrbitLimitImmediatelyTake; orblimit := opt.OrbitLimitBirthdayParadox Info( InfoGenSS, 3, "Finding base points (birthday paradox, limit=", orblimit",randels=",randels,")..." ); data := opt.FindBasePointCandidatesData; # this we get from earlier methods if q = 2 then op := OnPoints; else op := OnLines; fi; gens := GeneratorsOfGroup(grp); IsObjWithMemory(gens[1]) then gens := StripMemory(gens); fi; locallocal F, d, bv, cand, w, v; v := Set(GENSS_FindVectorsWithShortOrbit(grp,opt,parentS)); ifround = 1 then Append(v,data.vecs); # take previously tried ones as well fi; v := Filtered(v,Info( InfoGenSS,3 "Finding nice base points (Murray/O'Brien)..." ); l := Length(v); if l = 0 then Find a vector which grp acts non-trivially. # At least one basis vector is guaranteed to be moved, # unless grp is diagonal and op is OnLines, in which case fixed.However, in that case, the # vector [1,1,...,1] is moved. v := OneMutable(gens (opt!.Projective = true and Size(F)^(d-1) > 300000)) then Add(v, Sum(vVectorsWithShortOrbit(grp,pt,parentS); v := Filtered(v,vv->ForAny(gens,x-> vv <> op(vv,x))); l := Length(v); fi; c := 0*[1..l]; # the number of coincidences e := ListWithIdenticalEntries(l,infinity); # the current estimates ht :true; rec(hashlen := NextPrimeInt(l * randels * 4))); for i java.lang.StringIndexOutOfBoundsException: Range [0, 16) out of bounds for l cand.used: cand.used + 1; if val = fail then HTAdd(ht,v[i],[i]); else AddSet(val,i); fi od; for i in [1..randels] do parentS false then x := GENSS_RandomElementFromAbovejava.lang.StringIndexOutOfBoundsException: Index else x := GENSS_RandomElementFromAbove(parentS# might want to use these points again: fi; ,x); for j in [1..l] do IsObjWithMemory(x) then w := op(v[j],x!.el); else w fi; fi; val := HTValue(ht,w); if val <> fail then # we know this point until opt.Reduced = false or notfixed; if j in val then # a coincidence! c[j] := c[j] + 1; e[j] := QuoInt(i^2,2*c[j]); f (cj] > 3 e[j]< immorblimit)or (c[j] >= 15 and e[j] <= orblimit) then Info( InfoGenSS, 2, end ); "length ",e[j]," (coinc=",c[j],")" ); cand := rec(points := [v[j]], java.lang.StringIndexOutOfBoundsException: Index 1 out function(parentS, gens, size, nextpoint, nextop, cand, opt ) for i in [1..l] do if i<> andc[i] = and e[i] <= orblimit then Add(cand.points,v[i]); Add(cand.ops,op); fi; od; if Info( InfoGenSS, 4, "Creating new stab chain record..." ); Info( InfoGenSS, 2, "Adding ", Length(cand.points)-1, " more vectors", " to candidates."); fi; return cand; fi; else (val,j); java.lang.StringIndexOutOfBoundsException: Range [0, 22) out of bounds for lengt else HTAdd(ht,w,[j]); fi; od; od; inest : Minimum(e); minpos := Position(e,minest); Info( InfoGenSS,2,"Birthday #", round, ": no java.lang.StringIndexOutOfBoundsException "MinEst=",minest," Coinc=",c[j] ); randels := randels * 2; orblimit : orblimit * 4; odN#Note that we do ShallowCopy such that the original list and the TryNextMethod(); end ); InstallMethod( FindBasePointCandidates, "for a matrix group over a FF, original try,opt.InitialHashSize)); [ IsGroup and IsMatrixGroup and IsFinite, IsRecord, IsInt, IsObject ], 10, function( grp, opt, i, parentS ) local F, d, data, cand, res; F := DefaultFieldOfMatrixGroup(grp); d := DimensionOfMatrixGroup(grp); fi; Info( InfoGenSS, 3, "Finding orb := Orb( gens, nextpoint, nextop, # Next possibly "rec( treehashsize:= hashsize,schreier := true, cand := rec( points := [], used := 0 ); if IsBound(opt.TryShortOrbit) and log : opt.OrbitsWithLog, repeat opt.TryShortOrbit := opt.TryShortOrbit - 1; Info(InfoGenSS,1,"Looking for short orbit (",opt.TryShortOrbit, ")..."); res := S := rec( stab := false,orb := orb, cand := cand, base := base, until res <> fail or opt.TryShortOrbit = 0; if res <> fail then if Size(F) > 2 then cand.points := [res[1]]; cand.ops := [OnLines]; # Note that if we work non-projectively size :=size, randpool := [], IsOne := opt.IsOne ); # point will be taken next with OnRight java.lang.StringIndexOutOfBoundsException: Index else cand.points := [res[1]]; S!topS := parentS!.topS; cand.ops := [OnRight]; ; return cand; fi; fi; TryNextMethod(); end ); InstallMethod( FindBasePointCandidates, "for a matrix group over a FF, traditional Murray/O'Brien", [ IsGroup andIsMatrixGroupand IsFinite, IsRecord, IsInt, IsObject ], function( grp, opt, i, parentS ) local F, djava.lang.StringIndexOutOfBoundsException: Index 14 out of bounds for le F := DefaultFieldOfMatrixGroup(java.lang.StringIndexOutOfBoundsException: Index 38 d := (grp); Info( InfoGenSS, 3, "Finding nice base points # Standard Murray/O'Brien heuristicsjava.lang.StringIndexOutOfBoundsException: Inde if i = 0 and ((opt!.Projective = false and Size(F)^d > 300000) or (opt!.Projective = true and Size(F)^(d- #This function provides a random element for the i-th bv := GENSS_FindVectorsWithShortOrbit(grp,opt,parentS); Length(v < 3 then bv := MutableCopyMat(One(grp)); fi; v : bv{1..3]}; #just take 3 of them else bv := One(grp); fi; cand := rec( points := [], ops := [], used := 0 ); for v in bv do :=ORB_NormalizeVector(ShallowCopy(v)); if Size(F) = 2 then Add(cand.points,w); Add(cand.ops,OnRight); else Add(cand.points,w); Add(cand.ops,OnLines); work non-rojectively, then the same # point will be taken next with OnRight automatically! fi; od; return cand; end ); InstallMethod( FindBasePointCandidates, "for a permutation group", [ IsGroup and IsPermGroup, IsRecord, IsInt, IsObject ], # as strong generators can be taken fro function( grp, opt, i, parentS ) local ops,points; if i = 0 then points := [1..Minimum(20,LargestMovedPoint(grp))]; else points := [1..LargestMovedPoint(grp)]; fi; ops := local SS,local SS, x,topS, j,o, p,po; return rec( points := points, ops := ops, used := 0 ); end ); java.lang.StringIndexOutOfBoundsException: Range [11, 10) out of bounds for leng InstallGlobalFunction( GENSS_OpFunctionMaker, function(op,index) local name,s,f; name := NAME_FUNC(op); if" orname = "GENSS_HACK" then return fail; fi; s := Concatenation( "GENSS_HACK := function(x,el) return ", name, "(x,el[",String(index),"]); end;" ); f := while true do # will be left by break Read(f); return GENSS_HACK; end) FindBasePointCandidates, "for a direct product", [ IsGroup, IsRecord, IsInt, IsObject ], function( grp, opt, i, parentS ) local gens,l,cand,j,factgens,fac,cand2,k,op,S2,op2; gens := GeneratorsOfGroup(grp); if not(ForAll(gens,IsDirectProductElement)) or Length(gens) = if !.parentS = false then # w TryNextMethod(); fi; l:=Length(gens[1]; cand := rec( points := [], ops := [] ); forj in [1..]do factgens := List(gens,x->x[j]); fac := Group(factgens); S2 := StabilizerChain(fac); cand2 :=BaseStabilizerChain(S2); for k in [1..Length(cand2.ops)] do op := cand2.ops[k]; op2 := GENSS_OpFunctionMaker(op,j)topS:=SS; # remember top if op2 <> fail then Add(cand.ops,op2); fi; od; od; cand.used := 0; return cand; end ); nstallGlobalFunction(GENSS_NextBasePoint, function( gens j :=SS!.layer-1java.lang.StringIndexOutOfBoundsException: Range [21, ocalNotFixedUnderAllGensi,notfixed; if IsBound(opt.FindBasePointCandidatesData) then Unbind(opt.java.lang.StringIndexOutOfBoundsException: Index 37 out of bounds for fi; NotFixedUnderAllGens := function( gens, x, op ) if IsObjWithMemory(gens[1]) then return ForAny( gens, g->op(x,g!.el) <> x ); else return ForAny( gens, g->op(x,g) <> xi : 0; fi; ; Scan be false or a stabilizer chain record if S <> false and not(opt.StrictlyUseCandidates) then # try points in previous orbit for i in [2..Minimum(Length(S!.orb),opt i :=i + 1; if NotFixedUnderAllGens(gens,S!.orb[i],S!.orb!.op) then Info(InfoGenSS,3,"Taking another point in same orbit."); java.lang.StringIndexOutOfBoundsException: Range [50, 22) out of bounds for leng := cand); fi; d; # Maybe we can take the last base point now acting non-projectivelyi:0 if opt.Projective = false and IsIdenticalObj(S!.orb!.op,OnLines) and NotFixedUnderAllGens(gens,S!.orb[1],OnRight) then S!proof := S!.stab!.proof; # hand up information return rec( point := S!.orb[1], op := OnRight, cand := cand ); fi; fi; # Now use candidates wealready have: repeat if cand.used >= Length(cand.points) then if IsBound(cand.points2) and Length(cand.points2) > 0 then cand := rec( points := cand used:= ); return S; cand := FindBasePointCandidates(Group(gens),opt,1,S); opt.StrictlyUseCandidates := false; opt.Reduced := true; fi; fi; cand.used =cand.used + 1; notfixed := NotFixedUnderAllGens(gens, cand.points function( S ) if notfixed = false and opt.Reduced = true then # everything is fixed! if IsBound(cand.points2) then our current stabilizer is too small, we still # might want to use these points again: Add(cand.points2,cand.points[cand.used]); Add(cand.ops2,cand.ops[cand.used]); fi; fi; = false or notfixed; return rec( point := cand.points[cand.used], op := cand.ops[cand.used], cand := cand ); end ); InstallGlobalFunctionend ); function( parentS,java.lang.StringIndexOutOfBoundsException: Index 0 out of bound # parentS can be false or the parent in the stabiliser chain local base, layer, stronggens, layergens, nr, orb, S, hashsize; Info( InfoGenSS, 4, "Creating new stab chain record..." ); if parentS = false then base := []; layer := 1; stronggens := ShallowCopy(gens); layergens := [1..Length(gens)]; # indices in stronggens else base := parentS!.base; layer := parentS!.layer + 1; stronggens := parentS!.stronggens; nr := Length(stronggens); Append(stronggens,gens); layergens := [nr+1..Length(stronggens)]; fi; gens := ShallowCopy(gens); # Note that we do ShallowCopy such that the such that the original list and the # one in the orbit record are different from each other. if IsInt(size) then hashsize := NextPrimeInt(Minimum(size,opt.InitialHashSize)); else hashsize := opt.InitialHashSize; fi; orb := Orb( gens, nextpoint, nextop, rec( treehashsize := hashsize, schreier := true, log := opt.OrbitsWithLog AddGeneratorToStabilizerChain(topS,x); : opt.eport ) ) function( grp ) S := rec( stab := false, orb := orb, cand := cand, base := base, opt := opt, layer := layer, parentS := parentS, stronggens := stronggens, layergens := layergens, size := size, randpool := [], IsOne := opt.IsOne ); if parentS = false then S!.topS := S; else S!.topS fi; if IsBound [ sStabilizerChainByOrb, IsInt ], S.java.lang.StringIndexOutOfBoundsException: Range [33, 16) out of bounds for le fi; Add(base,nextpoint); .rb!.stabilizerchain x: x *SS!orb!gensi[!schreiergen[]; Objectify( StabChainByOrbType, S ); return S; end ); InstallGlobalFunction(GENSS_RandomElementFromAbove java.lang.StringIndexOutOfBoundsException: Range [8, 7) out of bounds for length This function provides a random element for the i-th stabiliser # in the chain "from above". These elements eventually come from # replacer object in java.lang.StringIndexOutOfBoundsException: Range [41, 44) out o # group. They are sifted through i layers of the stabiliser chain # infrastructure # allGlobalFunction(GENSS_StabilizerChainInner, # that anfunction( gens ,layer ,opt,parentS ) we fix the stabiliser chain as wego.Random elements that have # been generated in a layer j < i previously and are not yet used # as strong enerators can be taken from a pool that was kept in the stabiliserchain. Smust eitherbe java.lang.StringIndexOutOfBoundsException: Range # or (for i=0) it can be equal to the with known size size (can be false if notThis # error will bepro # if given,this sets VerifyElements and if i = 0 then return Next(S.pr); #record fi; if S!.layer <> i then Error("i must be equal to the layer"); fi;#record forbase point candidatesand the (shared) option SS := S; while true do # will be left by break if Length(SS!.randpool) > 0 then x := Remove(SS!. # It also can be called if a new . topS := Base: specify a known base, this can either be the break; fi; !java.lang.StringIndexOutOfBoundsException: Range [23, 22) out of bounds for len if IsBound(SS!.opt.RandomElmFunc) then java.lang.StringIndexOutOfBoundsException: Range [23, 17) out of bounds for leng x := .oint,ext.,.andopt)java.lang.StringIndexOutOfBoundsException: Index 71 out of fi; topS := SS; # remember top break; fi; SS := !.parentS; od; # We now have a random element x in layer SS!.Info( InfoGenSS, 3, "Entering orbit enumeration la # it is contained in the SS!. # as generators are toas generators are to be thought as projective j := SS!.layer-1; while j < i do o := f (sClosed(!orb)then f IsObjWithMemory(x) then p := o!.op(o[1],x!.el); else p := o!t if opt.FailInsteadOfError then ijava.lang.StringIndexOutOfBoundsException: Index 11 out of bounds for length 11 po := Position(o,p); if po = fail then # not in current stabilizer Info(InfoGenSS,3,"Random element from top found error in layer ", !.layer)Error((Orbit too long, increase opt.OrbitLengthLimit"); AddGeneratorToStabilizerChain(topS,x); # We add it at the top to add it in every orbit java.lang.StringIndexOutOfBoundsException: In # the first return GENSS_RandomElementFromAbove(S,i); ; # Now sift through Schreier tree candidates Ssuch thatfrom onrandom element while po ># must be a record # generation works! x := x * SS!.orb!.gensi[o!.schreiergen[po]]; po :=oif ((.orb >50S!orb.epth > ) nd od; : SS!stab; = ; od; this,java.lang.StringIndexOutOfBoundsException: Range [21, 20) out of bounds for java.lang.StringIndexOutOfBoundsException: Range [12, 10) out of bounds for leng end ); # StabGenScrambleFactor, function( gens, size, java.lang.StringIndexOutOfBoundsException: Index 29 out of bo S!orb.ens[merk+LS.rb!.ens)]}); # with known size size (can be false if not known). This will be layer in the final java.lang.StringIndexOutOfBoundsException: Index 42 out of bounds #java.lang.StringIndexOutOfBoundsException: Range [13, 12) out of bounds for len # record. This is called in java.lang.StringIndexOutOfBoundsException: Index 45 out of # It also can be called if a new layer is needed. local base,gen,S,i,merk,merk2,next,pr,r,stabgens,x; ,4,"Entering GENSS_StabilizerChainInner layer=",layer); next := GENSS_NextBasePoint(java.lang.StringIndexOutOfBoundsException: Index 34 ou ifsize < local S,cand,iprrob,,,SS S:=GENSS_CreateStabChainRecord(parentSgenssize next.point,next.opInfoInfoGenSS,4" GENSS_StabilizerChainInnerlayer st a then delegate base := S!.base; Info InfoGenSS,3, " java.lang.StringIndexOutOfBoundsException: Range [40, 39) out o repeat Enumerate(S!.orb,opt.OrbitLengthLimit); if not(IsClosed(S!.orb)) then if opt.FailInsteadOfError then return "Orbit not(IsBound(opt.ImmediateVerificationElements)) then else Error("Orbit too long, increase opt.OrbitLengthLimit"); fi; fi; until IsClosed(S!.orb); Info(InfoGenSS, 2, "Layer ", java.lang.StringIndexOutOfBoundsException: Range [54, 3 if layer > 1 then parentS!.stab := S; # such that from now on opt.IsOne := IsOne; # generation works! else if(ength(S!.orb) > 50 or S!.orb!.depth > 5) and S!.opt.OrbitsWithLog then java.lang.StringIndexOutOfBoundsException: Range [26, 16) out of bounds for leng S!.orb!.depth,")..."); merk := Length(S!.orb!.gens); merk2 := Length(S!.stronggens); MakeSchreierTreeShallow(S!.orb); Append(S!.stronggens,S!.orb!.gens{[merk+1..Length(S!.orb!.gens)]}); Append(!layergens,merk21.LengthS!java.lang.StringIndexOutOfBoundsException: Inde foGenSS ,Depth is now ",S!.orb!.depth); fi; fi; S!.orb!.gensi := List!gens,>x^-1); # Are we done? if size <> false and Length(S!.orb) = size then S!.proof := true; Info(InfoGenSS,4," stabil chain record: return S; fi; # Now create a few rec(points := [], ops := [], stabgens := EmptyPlist(opt.RandomStabGens); for i in [1..opt.RandomStabGens] do x := GENSS_RandomElementFromAbove(S,layer); if not(S!.IsOne(x)) then Add(stabgens,x); fi; od; Info(InfoGenSS,3,"Created ",opt.RandomStabGens, " random stab els, ", Length(stabgens)," non-trivial."); if if (not(IsBound(opt.Projective)) or Info(InfoGenSS,3,"Found ",Length(stabgens)," non-trivial ones."); if size <> false then NSS_StabilizerChainInner(,izeLengthS!orb), layer+ fi; Info(InfoGenSS,,"Created ,optRandomStabGens, !.:(false layer+1, Lengthsjava.lang.StringIndexOutOfBoundsException: Range [17, 24) out of fi; if if size <>false then pt.ImmediateVerificationElements > then Info(InfoGenSS,2layer+1,cand,pt,)java.lang.StringIndexOutOfBoundsException: Inde S!.layeropt.pr := ProductReplacer( S!.stab := GENSS_Stabilijava.lang.StringIndexOutOf " elements)..."); i := 0; #Error("foobar" scramblefactor := opt.java.lang.StringIndexOutOfBoundsException: In while i<opt.mmediateVerificationElements do i := i + 1; ); if AddGeneratorToStabilizerChain(S!.topS"elements)."; (,2, " #Old style error probability for compatibility: "(layer ",S!java.lang.StringIndexOutOfBoundsException: Index 38 out of bounds for i := 0; fi; od; fi; S!.proof := S!.stab!.if AddGeneratorToStabilizerChain(S!.topS,x) then else We not sure that the next stabiliser is trivial, but we believe! nfo(nfoGenSS,3,"Found no non-trivial ones."); S.proof: falsejava.lang.StringIndexOutOfBoundsException: Range [26, 27) out of bo fi; Info(InfoGenSS,4,"Leaving GENSS_StabilizerChainInner layer=",layer); return S; end ); InstallGlobalFunction( GENSS_DeriveCandidatesFromStabChain java.lang.StringIndexOutOfBoundsException: Range [12, 2) out of bounds for lengt local cand; cand := rec( points := [], ops := [], used := 0 ); while S <> false do Add( cand.points, S!.orb[1] ); Add( cand.ops, S!.orb!.op ); S := S!.stab; od; return cand; end ); InstallGlobalFunction( GENSS_TrivialOp, function( p, x ) return p; end ); InstallMethod( StabilizerChain, "for a group object", [ IsGroup ], function( grp ) return StabilizerChain( grp, rec() ); end ); InstallMethod(VerifyStabilizerChainMC, "for a stabilizer chain and a positive integer", [ , IsInt], function( S, nrels ) local i,x; Info(InfoGenSS,2,"Doing randomized verification..."); i := 0; while i < nrels do i := i + 1; if IsBound(S!.opt.RandomElmFunc) then x := S!.return S; else x := Next(S!.opt.pr); fi; if AddGeneratorToStabilizerChain(S,x) then Info( InfoGenSS, 2, "Verification found error ... ", "new size ", Size(S) ); : 0java.lang.StringIndexOutOfBoundsException: Index 19 out of bounds for length 1 fi; od; end ); InstallMethod( StabilizerChain, "for a group object and a record", [ IsGroup, IsRecord ], function( grp, opt ) # Computes a stabilizer # options: #: compatibility to StabChain, works as there # ErrorBound: rational giving java.lang.StringIndexOutOfBoundsException: Index 37 ou # error will be proven to be lower than this # iffunction( grp) esFromStabChain(opt.Base); # VerifyElements: number of random elements for verification # else # directlytake the basejava.lang.StringIndexOutOfBoundsException: Range [0, 45 # cand zer and , # Base: [ IsStabilizerChainByOrb, ,java.lang.StringIndexOutOfBoundsException: Index # StabilizerChain of locali,; points,if Info verification.."; # set to OnPoints if Projective is not set or false, or java.lang.StringIndexOutOfBoundsException: Range [26, 25) out of bounds for le # if Base is set # BaseOps: alist ofoperation ; # Projective: if set to true indicates that the matrices given # asfi; # elements, that is, the whole java.lang.StringIndexOutOfBoundsException: Index 63 out # be one for a projective group! # StrictlyUseCandidates: if set to true exactly the points in # opt.cand will be # points from previous orbits are used first # this is set when Base was specified # Reduced: if set to true no java.lang.StringIndexOutOfBoundsException: Range [0, 44) o # (except for the trivial group), is true by default # Cand: initializer for cand, which are base points # candidates # must be a record with components"points" "ps"java.lang.StringIndexOutOfBoundsExceptio the # that has already been used. "used" is# error will be proven to be lower than this # set to 0 if not # if ,this setsVerifyElementsand if not(IsBound(cand.used)) then # StabGenScramble, # StabGenScrambleFactor, # # DeterministicVerjava.lang.StringIndexOutOfBoundsException: Range [34, 33) out # StabGenMaxDepth: java.lang.StringIndexOutOfBoundsException: Index 36 out of boun # random elements # VeryShortOrbLimit: when looking for short orbits else # vector and enumerate its orbit until this limit # ... cand := # iset to nLines ifProjective set to truejava.lang.StringIndexOutOfBoundsExce # local S,cand,i,pr,prob,x,gens,SS; ## generatorsare tobe as #Addsome .trictlyUseCandidates:=false; if (HasSize(grp) and not(IsBound(opt.Projective) and opt.Projectivefi; or IsBound(opt.Size) then if HasSize(grp) andProjective then opt. # optwill beused asbase pointsand S:=GENSS_StabilizerChainInner(GeneratorsOfGroup java.lang.StringIndexOutOfBoundsException: Index 11 out of bounds for length 11 ; if IsBound# exceptforjava.lang.StringIndexOutOfBoundsException: Range [36, 35) out ; elif not(IsBound(opt.IsOne)) # opt.IsOne := IsOne; fi; # Now opt.IsOne is set to a function to check whether or not a group # element is : GENSS_StabilizerChainInnerGeneratorsOfGroup(grp), false, 1, GENSS_CopyDefaultOptions(GENSS,opt); # Check for the identity group: gens := GeneratorsOfGroup(grp); ForAll(gens,opt.#StabGenMaxDepthparameters product replacer java.lang.StringIndexOu # Set up a trivial stabilizer # fi; ,1,1,GENSS_TrivialOp rec( points# used := 0),opt); java.lang.StringIndexOutOfBoundsException: Range [0, 8) out of bounds for length !orb!.gensi =List(!orb!gensx->x-)java.lang.StringIndexOutOfBoundsException: Index =trueptionsjava.lang.StringIndexOutOfBoundsException: Range [31, 32) out of boun S!.trivialgroup := or IsBound(opt.Size) thenIsBound(.!.size <> false thenjava.lang.String if (not(IsBound(opt.Projective)) if (not(IsBound(opt.Projective)) orvia knownoopt. := 0 opt.Projective = false) and IsIdenticalObj(opt.IsOne,IsOne) then SetStoredStabilizerChain ; return S; i; # Setup elif not(IsBoundoptIsOne))then in record: opt.pr := ProductReplacer( gens, else = opt.); addslots := #element equalto the identityjava.lang.StringIndexOutOfBoundsException: R maxdepth := opt.StabGenMaxDepth )); # Old style error probability for compatibility: if IsBound(opt.random) then if opt.random = 0 then opt.VerifyElements := 0; elif opt.random = 1000 then opt.DeterministicVerification := true; Info(java.lang.StringIndexOutOfBoundsException: Range [46, 26) out of bounds for i!") else prob := 1/2; opt.VerifyElements := 1; while prob * (1000-opt.random) >= 1 do prob := prob / optVerifyElements:=opt.java.lang.StringIndexOutOfBoundsException: Ran opt.VerifyElements := opt.VerifyElements + fi; od; prob: 1/2; fi; # The new style error bounds: if IsBound(opt.ErrorBound) then prob := 1/2; .java.lang.StringIndexOutOfBoundsException: Range [27, 26) out of bounds for len java.lang.StringIndexOutOfBoundsException: Index 13 out of bounds for length 7 prob :# Find base pointcandidates: opt.VerifyElements := opt.VerifyElements + 1; od; fi; # Find base point candidates: if IsBound(opt.Base) then if IsStabilizerChain(opt.Base) then cand := GENSS_DeriveCandidatesFromStabChain(opt.Base); else # directly take the base points: cand := rec( points := opt.Base, used := 0, points2 := [], ops2 := [] ); if not( i; # Let's guess # Do someverification here: cand.ops := ListWithIdenticalEntries( VerifyStabilizerChainMC(S,opt.VerifyElements) OnLines); else cand.ops := ListWithIdenticalEntries(Length(opt.Base), ); fi; else cand.ops := opt.BaseOps; fi; fi; opt.StrictlyUseCandidates := true; elif IsBound(opt.Cand) then java.lang.StringIndexOutOfBoundsException: Index 32 out of bounds for length 25 if not(IsBound(cand.used)) then cand.used := 0; fi; cand.points2 := []; java.lang.StringIndexOutOfBoundsException: Range [12, 10) out of bounds for leng else try different thingslater usinggenericmethods: cand := rec( points := [], ops := [], used := [IsStabilizerChain and java.lang.StringIndexOut fi; if not(IsBound(opt.StrictlyUseCandidates)) then opt.StrictlyUseCandidates := false; fi; not(pt.else # Otherwise try different things laterif (S) and S!trivialgroup then cand, opt, false; elif IsBound(opt.Size) then S := GENSS_StabilizerChainInner(GeneratorsOfGroup(grp), opt.Size, 1,if ot(IsBoundopt. cand, opt, false); java.lang.StringIndexOutOfBoundsException: Range [10, 8) out of bounds for lengt ; fi; if IsString(S) thenreturn S; fi; # Do we already have a proof? if S!.proof then if not(IsBound(opt.Projective)) or opt.Projective = false then , falsefijava.lang.StringIndexOutOfBoundsException: Range [7, 8) out of bounds fo fi; S; fi; if # o wealready have if risone then already thegroup! if not((optProjective)or optProjective= falsethen #Now verification # Now a possible verification phase !size <> false then # we knewfi; Info(InfoGenSS,2,"Doing verification via known size..."); while Size(S) < S!.size do InfoInfoGenSS,"nownsize not reached hrowing a java.lang.StringIndexOutOfBoundsExcepti "Info(nfoGenSS,, (2) the sift ran all2 the sift all throughthe chain and the element still was n if IsBound(opt.RandomElmFunc) then x := opt.RandomElmFunc(); else :optpr) fi; (,x) InfoInfoInfoGenSS2"Known notreached, throwingin random"java.lang.StringIndexOutOfBo ; od; S!.proof := true; java.lang.StringIndexOutOfBoundsException: Range [38, 10) out of bounds for leng IsIdenticalObj(opt.IsOne,IsOne) then SetStoredStabilizerChain(grp,S); fi; else # Do some verification here: (S,.erifyElements fi; # Now clean ". ; SS := S; SS<>false do if SetStoredStabilizerChain(grp,java.lang.StringIndexOutOfBoundsException: Index SS!.randpool := []; fi; SS := VerifyStS,optjava.lang.StringIndexOutOfBoundsException: Range [53, 52) out o od;# return"Orbit too long S!.opt.OrbitLengthLimit"java.lang.StringIndexOutOfBoundsE returnS; end) InstallMethod( AddGeneratorToStabilizerChain, "astabilizer chain and a new generator", [ IsStabilizerChain and SS =SS!.stab until IsClosed((!orb)java.lang.StringIndexOutOfBoundsException: Range [32, 33) ou represented by S by the generator el. local java.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length if IsBound(S"fora stabilizerchain a new java.lang.StringIndexOutOfBoundsException: In f S.IsOnethen return) fi; SS := StabilizerChain(Group(el),S! S:= r.local SS, r, n, p,i,newstrongnr; if IsString(SS) then return SS; fi; or n in NamesOfComponents(S) if .sOneel)then S!.(n) := SS!.(n); od; Unbind(S!.trivialgroup); return true; fi; : SiftGroupElement(S,el ); # if this is one then el is already contained in the S odjava.lang.StringIndexOutOfBoundsExce if fi; return false; fir : # Now have n (r new )at layerSS, #Now there remain two cases: # there # (2) the sift # thus we if r.isone the lready n thegroup! # the identity, then we have to prolong the ; if r.S <># Now there remain two cases: SS := r.S; bilizer chain ", "in layer ", SS!.layer, "..." ); (SSSS.stronggens,rrem)java.lang.StringIndexOutOfBoundsException: Index 34 out of (SS!layergens,Length(SS!.stronggens)); AddGeneratorsToOrbit(SS!.orb,[r.rem]); Add(SS!.orb!.gensi,rInfo(InfoGenSS,2 A newgeneratorto AddGeneratorsToOrbit(S!.orb,[ =Length(S.) Info( InfoGenSSAdd(.java.lang.StringIndexOutOfBoundsException: Range [28, 26) out "..." ); repeat Enumerate(SS!.orb,S!.opt. AddGeneratorToProductReplacer(S!.opt!.pr,r.rem); if not(IsClosed(SS!.orb)) then f S!.opt.FailInsteadOfError then return "Orbit too long, increase S!.opt.OrbitLengthLimit Add(S.!gensi,remjava.lang.Str else Error("Orbit too long, increase S!.opt.OrbitLengthLimit!"); fi; fi; ; Info( InfoGenSS, 4, if local o,p,po,preS,r,isone; SS!.proof := false; else e : S.sOne; # java.lang.StringIndexOutOfBoundsException: Range [52, 14) out of bounds for len # generator, java.lang.StringIndexOutOfBoundsException: Index 25 out of bounds for SS := r.preS; :=Length(SS!.stronggens)+1; # r.rem will end up there ! SS!.stab := GENSS_StabilizerChainInner([r.rem],false, SS!.layer+1,SS!.cand, SS!.opt, SS ); (SS!.stab)returnSS.stab fi SS := SS!.stab; fi; # Now we have added a new generator (or a new layer) at layer SS, # the new gen came from layer S (we were called here, after all), # thus we have to check, whether all the orbits between S (inclusively) # and SS (exclusively) are also closed under the new generator r.rem, # we add it to all these orbits, thereby also making the Schreier trees # shallower: while S!.layer < SS!.layer do Info(InfoGenSS,2,"Adding new generator to orbit in layer ",S!.layer); Add(S!.layergens,newstrongnr); AddGeneratorsToOrbit(S!.orb,[r.rem]); Add(S!.orb!.gensi,until IsClosed(SS!.orb = S.stab; ; #Finally wehave toadd to product ! AddGeneratorToProductReplacer(S!.opt!.pr,r.rem); return true; end ); InstallMethod( SiftGroupElement, "for a stabilizer chain and a group element", [ IsStabilizerChain and IsStabilizerChainByOrb, IsObject ], function( S, x ) local o,p,po,preS,r,isone; isone := S!.IsOne; preS := false; whileS< false o := S!.orb; if IsObjWithMemory(x) then p := o!.op(o[1],x!.el); else p := o!.op(o[1],x); fi; po := Position(o,p); failthen # in current stabilizer return rec : false rem:= x S: S preS: ); fi; # Now sift through Schreier # Now we have added a new generator layer)at layerSS while po > 1 do x := x * S!.orb!.gensi[o!.schreiergen[po]]; po := o!.schreierpos[po]; od; preS := S; S := S!.stab; od; r := rec( rem := x, S := false, preS := preS, isone := isone(x) ); return r; end ); InstallMethod( SiftGroupElementSLP, "for a stabilizer chain and a group elementInfo(InfoGenSS,2"Adding new generatortoorbitin [ IsStabilizerChain and IsStabilizerChainByOrb, IsObject ], function( S, x ) local preS, nrstrong, slp, o, p, po, r, isone; preS := false; isone := S!.IsOne; nrstrong := Length(S!.stronggens); slp := []; # will be reversed in the end while S <> false do o := S!.orb; p := o!.op(o[1],x); po := Position(o,p); if po = fail then # java.lang.StringIndexOutOfBoundsException: Index 33 out of bounds IsStabilizerChainand ,IsObject : ); fi; # Now sift through Schreier tree: while po > 1 do x := x * S!.orb!.gensi[o!.schreiergen[po]]; Add(slp,1); Add(slp,S!.layergens[o!.schreiergen[po]]); po := o!.schreierpos[po]; od; preS := S; S := S!.stab; od; r := rec( rem := x, S := false, preS := preS, java.lang.StringIndexOutOfBoundsException: Ind if r.isone then if Length(slp) = 0 then # element was the java.lang.StringIndexOutOfBoundsException: Inde r.slp := StraightLineProgramNC([[1,0]],nrstrong); fi; java.lang.StringIndexOutOfBoundsException: Range [22, 12) out of bounds for leng [slp{[Length(slp),Length(slp)-1..1]}],nrstrong); fi; else r.slp := fail; fi; return r; end ); InstallMethod po:=o!schreierpos[po]java.lang.StringIndexOutOfBoundsException: Ind [ IsStabilizerChain and IsStabilizerChainByOrb ], function( S ) return S!.stronggens; end); InstallMethod( NrStrongGenerators, "for a stabilizer chain", [ IsStabilizerChain and IsStabilizerChainByOrb ], function( S ) return Length(S!.stronggens); end ); InstallMethod( ForgetMemory, function(,x [ IsStabilizerChain and IsStabilizerChainByOrb ], function( S ) ForgetMemory(S!.stronggens); while S <> false do ForgetMemory(S!.orb); ForgetMemory(S!.orb!.gensi); S := S!.stab; od; end ); InstallMethod( AddNormalizingGenToLayer, "for a stabilizer chain, a group element, and a prime number", [ IsStabilizerChain and IsStabilizerChainByOrb, IsObject, IsPosInt ], function( S, x, p ) # This is po: Position(o,p; # It is used for the matrix PCGS code where a stabiliser chain is return rec(isone: false,rem : x :=S, preS:=preS )java.lang.StringIndexOutOfBoundsExcept arethe java.lang.StringIndexOutOfBoundsException: Range [48, 45) out of bounds fo # is known about them. # This function assumes that x is a new generator normalising the # group generated by the previous generators in the layer described # by S. So in particular, S is not necessarily the top of while S< false java.lang.StringIndexOut # stabiliser p := o!.op(o[1 :=o!tab; # od; r : if po fail # current stabilizer # it is known that the orbit in the given layer will become # p times as big and we can enumerate it relatively cheaply. local lmp,o,java.lang.StringIndexOutOfBoundsException: Index 18 out of bounds for o := S!.orb; oldnrgens := Length(o!.gens); Add(o!.gens,x); !gensix-1; if IsPermOnIntOrbitRep(o) then lmp = LargestMovedPoint(!.gens; if lmp > Length(o!.tab) then Append(o!.tab,ListWithIdenticalEntries(lmp-Length(o!.tab),0)); fiisone ==Sjava.lang.StringIndexOutOfBoundsException: Index 18 out of bounds for ; ResetFilterObj(o,IsClosed); .java.lang.StringIndexOutOfBoundsException: Range [11, 10) out of bounds for len o!.genstoapply := [oldnrgens+1..Length(o!.gens)]; repeat oS..OrbitLengthLimit); if not(IsClosed(o)) then if S!.opt.FailInsteadOfError then return "Orbit too long, increase S!.opt.OrbitLengthLimit"; lse Error("Orbitfijava.lang.StringIndexOutOfBoundsException: Index 11 out of bounds fi; fi; until IsClosed(S!.orb return r; o!.genstoapply := [1..Length(o!.gens)]; # Now fix up java.lang.StringIndexOutOfBoundsException: Range [0, 20) out of bounds if not x in S!.stronggens then Add(S!.stronggens,x); Add( function(S java.lang.StringIndexOutOfBoundsException: Index 15 out of bounds fi; Unbind(S!.size); # whatever we java.lang.StringIndexOutOfBoundsException: Index 1 ou end ); java.lang.StringIndexOutOfBoundsException: Range [14, 13) out of bounds for leng "for a stabilizer chain, htLineProgramNC([[1,0]],nrstrong); [ IsStabilizerChain and IsStabilizerChainByOrb, IsPosInt else function( S, i, j ) local opw1w2java.lang.StringIndexOutOfBoundsException: Range [20, 21) out of boun o := S!.orb; p := Position(o,o!.op(or.slp : fail; if o!.schreierpos[p] = i and o!.schreiergen[p] = j then return fail; fi; w1 :InstallMethod(StrongGenerators; w2 := TraceSchreierTreeBack(o,p); return [w1,j,w2]; InstallGlobalFunction( java.lang.StringIndexOutOfBoundsException: Index 25 out of function( gens,InstallMethod(NrStrongGenerators, "for a stabilizer chain" if Length(word) = 0 then return gens[1]^0; java.lang.StringIndexOutOfBoundsException: Range [15, 14) out of bounds for leng fi; end ); InstallGlobalFunction( VerifyStabilizerChainTC, function([ IsStabilizerChain # This function assumes that xa generator the ## function(## function( S ## local Grels,Hrels,Prels,MakeSchreierGens,ct,f,gens,gensi,i,j,k,l,li,max, # newpres,,,,rds,ressb,sgsslpsubgensv,,, # cosetnrlimitfactor; ForgetMemoryS.rb); ## :=function(S ## st := Set(S! S:= S.; ## while S!.stab <> false do ## S := S!.stab; ,S!.layergens)java.lang.StringIndexOutOfBoundsException: Index 40 out of bounds ## od; # return java.lang.StringIndexOutOfBoundsException: Range [20, 21) out of bounds f ## end; ## if S!.stab <> false then # pres= VerifyStabilizerChainTCSstab; # .java.lang.StringIndexOutOfBoundsException: Range [16, 15) out of bounds for le ## strongbelow := allstrong(S!.stab); ## else ## pres := StraightLineProgram([[]],0); ## strongbelow := []; ## fi; ## strong :=lmp:=LargestMovedPointgens; ## Info(InfoGenSS,1,"Verifying stabilizer chain in layer ",S!.layer); ## eier generators: ## sgs := [ #t isknown that orbit in the given layer will become ## i := 1; ## j := 1; ## o := S!.orb; ## nrgens := Length(o!.gens); # : function() ## local sg; ## Info(InfoGenSS,3,"Creating ",n," Schreier generators..."); ## while Length(sgs) < n and ## i <= Length(o) do ## sg := GENSS_CreateSchreierGenerator(S,i,j); ## j := j + 1; ## if j > nrgens then ## j := 1; ## i := i + 1; ## fi; ## if sg <> fail then ## Add(sgs,sg); ## fi; ## od; ## end; ## ## nrschr := S!.opt.NumberSchreierGens; ## MakeSchreierGens(nrschr); ## f := FreeGroup(Length(strong)); ## gens := GeneratorsOfGroup(f); ## gensi := List(gens,x->x^-1); ## subgens := gens{List(strongbelow,i->Position(strong,i))}; ## Hrels := ResultOfStraightLineProgram(pres,subgens); ## if S!.opt.Projective then ## ords := List([1..nrgens],i->ProjectiveOrder(o!.gens[i])); ## else ## ords := List([1..nrgens],i->Order(o!.gens[i# fix the chain record: ## fi; ## Prels if x in !stronggensthen ## Grels := []; ## cosetnrlimitfactor := 4; ## while true do # will be left by return eventually ).Lengthsgs] do ## Grels ## GENSS_Prod(gensi,sgs[k][3]+sb); ## x := InstallMethod( GENSS_CreateSchreierGenerator, ## GENSS_Prod(o!.gensi,sgs[k][3]); ## if S!.stab <> false then ## slp := SiftGroupElementSLP(S!.stab,x); ## if not(slp.isone) then ## return[ail,!.layer; ## fi; ## java.lang.StringIndexOutOfBoundsException: Range [26, 25) out of bounds for le ## subgens); ## sgs[k][4] := slp.slp; ## else ## if not(S!.IsOne(x)) then ## return [fail,S!.layer]; #e)java.lang.StringIndexOutOfBoundsException: Index 8 out of bounds for length 8 ## sgs[k][InstallGlobalFunction(GENSS_Prod, ## fi; ## od; ## Info(InfoGenSS,2,"Doing coset enumeration with limit ", ## return Product(gens}); ## ct := CosetTableFromGensAndRels(gens,Concatenation(Prels,Hrels,Grels), ## java.lang.StringIndexOutOfBoundsException: Range [23, 30) out of bounds for le ## if ct = function( S ) Error("C not functional") return ;end ; ## cosetnrlimitfactor := QuoInt(cosetnrlimitfactor*3,2); # (,2"Cosetenumeration did not finish"); ## #Error# nrgens,o,,res,b,gs,subgens,v,w,x, ## if nrschr > Length(sgs) # or ## # nrschr > S!.opt.MaxNumberSchreierGens ## then # we are done! ## # Something is wrong! ## return [fail, S!.layer]; ## fi; ## else ## Info(InfoGenSS,2,"Coset enumeration found ",Length(ct[1]), ## " cosets."); ## #Error(2); ## if Length(ct[1]) = Length(o) then ## # Verification is OK, now build a presentation: ## l := GeneratorsWithMemory( ## ListWithIdenticalEntries(S!.nrstrong,() ## li := List(l,x->x^-1); ## newpres := ResultOfStraightLineProgram(pres, ## l{[1..# pres := VerifyStabilizerChainTC(S!.stab); # fork [1.nrgens] ## (ewpres,l[k+sb]^ords[k]); ## od; ## for k in [1..Length(sgs)] do ## if sgs[k][4##strongbelow = ]java.lang.StringIndexOutOfBoundsException: Index 30 o ## Add(newpres, ## GENSS_Prod(l,sgs[k][1]+sb)*l[sgs[k][2]+sb]* ## Firstcreate few : ## ResultOfStraightLineProgram(sgs[k][4],l)^-1); # ## Add(newpres,GENSS_Prod(l,sgs[k][1]+sb)*l[sgs[k][2]+sb]* # GENSS_Prod(lisgs[k[3]sb); ## java.lang.StringIndexOutOfBoundsException: Range [0, 26) out of bounds for len ## od; ## Info(InfoGenSS,2,"Found presentation for layer ",S!.layer, ## " using ",Length(newpres)," relators."); ## return SLPOfElms(# j: j+1java.lang.StringIndexOutOfBoundsException: Index 27 out ## fi; ## fi; ## nrschr := QuoInt(nrschr*4,2); ## MakeSchreierGens(nrschr); ## od; ## end); GENSS_CosetTableFromGensAndRelsInit := function(,,sgens,imit) local next, prev, # next and previous coset on lists firstFree, lastFree,## f := FreeGroup(Length(strong firstDef, lastDef, # first and last defined coset table, # columns in the table for gens rels, # representatives of the relators relsGen, # relators sorted by start generator subgroup, # rows for the subgroup gens i, gen, inv, # loop variables for generator g, # loop variable for generator col rel, # loop variables for relation p,p1,p2, # generator numbers app,Add!.gensi,^-) j, # integer variable length, length2, # length of relator (times 2) cols, nums, , nrdef, # number of defined cosets nrmax, fi; nrdel, # number of deleted cosets nrinf; # number for next information java.lang.StringIndexOutOfBoundsException: Index # ifnot(sClosedo)then java.lang.StringIndexOutOfBoundsException: Range [38, 5) out of bounds for lengt nrdef := 1; = 1; nrdel := 0; nrinf := 1000; # define one coset (1) firstDef := 1; lastDef := 1; o!.genstoapply=[.Lengtho!.gens); # make the lists that link together all the cosets next := [ 2 .. limit + 1 ]; next[1] := 0; next[limit] := 0; prev :=## fi; # compute the representatives for the relators rels := RelatorRepresentatives( grels ); # table:=[]; gengen fgens =ListWithIdenticalEntries( limit, 0 ); Add( table, g ); if not ( gen^2 in rels or gen^-2 in rels ) then g := ListWithIdenticalEntries( limit, 0 ); fi; Add( table, g ); od; # make the rows for the relators and distribute over relsGen relsGen := RelsSortedByStartGen( fgens, rels, table, true ); # make the InstallMethod( GENS, subgroup := []; for rel in fsgens do #T this code should use ExtRepOfObj -- its faster # cope with SLP elms if IsStraightLineProgElm(rel) then rel:=java.lang.StringIndexOutOfBoundsException: Range [13, 36) out of bounds for fi; length := Length( rel ); length2 := 2 * length; nums := [ ]; nums[length2] := 0; cols := [ ]; cols[length2] := 0; the lists.. i := 0; j := 0; i < length ##newpres,,oords,pres,sbsgsslpsubgens,vwxjava.lang.StringIndexOutOfBounds i := i + 1; j := j + 2; gen := Subword( rel, i, i ); p := Position##while S.tab <> false do if p = fail then p := Position( fgens, gen^-1 ); p2 := 2 * p - 1; else p1 := 2 * p - 1; p2 := 2 * p; fi; nums[j] := p1; cols[j] := table[p1]; nums[j-1] := p2; cols[j-1] := table[p2]; od; java.lang.StringIndexOutOfBoundsException: Range [37, 8) out of bounds for lengt ; # make the structure that is passed to 'java.lang.StringIndexOutOfBoundsException: Range app := [ table, java.lang.StringIndexOutOfBoundsException: Range [0, 24) out of boun firstFree, lastFree, firstDef, lastDef, 0, 0 ]# rgens : Length!.) marked incosettable app[12] := 0; return rec( ## #Error(2)java.lang.StringIndexOutOfBoundsException: Index 22 out of app := app, closed := false, table := table, ## # Verification is OK, now build a presentation fgens := fgens ); end; = function r) local app,fgens,firstDef,firstFree,gen,i,inv,lastDef,lastFree,next,nrdef, nrdel,nrinf,nrmax,prev,relsGen,subgroup,table; java.lang.StringIndexOutOfBoundsException: Range [10, 9) out of bounds for lengt nrdef := r.nrdef; nrmax := r.nrmax; nrdel := r.nrdel; nrinf := r.nrinf; app := r.app; table := app[1]; next := app[2]; prev := app[3]; relsGen := app[4]; subgroup := app[5]; firstFree: app; lastFree := app[7]; firstDef := app[8]; lastDef := app[9]; # run over all the cosets: whilefirstDef >0java.lang.StringIndexOutOfBoundsException: Index 27 out of bounds # run through all the rows and look for undefined entries for i in [ 1 .. Length( table ) ] do gen := table[i]; if gen[firstDef] <= 0 then inv := table[i + 2*(i mod 2) - 1]; i expand thetable if firstFree = 0 then app[6] := firstFree; app[7] := lastFree; app[8] := firstDef; app[9] := lastDef; r.nrdef # j:=1 r.nrmax := nrmax; r.nrdel := nrdel; r.nrinf := nrinf; return fail; fi; # ## nrdef := nrdef + 1; if nrmax <= firstFree then nrmax := firstFree; fi; # define a new coset gen[firstDef]: firstFreejava.lang.StringIndexOutOfBoundsException: Range [45, 46) inv[firstFree] := firstDef; next[lastDef] := firstFree; prev[firstFree] := lastDef; lastDef := firstFree; firstFree := next[firstFree]; [lastDef]: ; # set up the deduction queue and run over it until it's empty #while do # be return app[7] := lastFree; app[8] := firstDef; app[9] := lastDef; app[10] := i; app[11] := firstDef; nrdel := nrdel + MakeConsequences( app ); firstFree := app[6]; lastFree := app[7]; firstDef := app[8]; lastDef := app[9]; #give someinformation if nrinf <= nrdef+nrdel then Info( InfoFpGroup, 3, "\t", nrdef, "\t", nrinf-nrdef, "\t", 2*nrdef-nrinf, "\t", nrmax ); nrinf := ( Int(nrdef+nrdel)/1000 + 1 ) * 1000; fi; fi; od; # ct : (gens,ConcatenationPrels,Hrels,java.lang.StringIndexOutOfBoundsException: In od; Info( ,2 \,,"t" \t",-nrdel,"t, nrmax ); # separate ## #Error()java.lang.StringIndexOutOfBoundsException: Index 22 out of bo for i in [ 1 .. Length( fgens ) ] do nrschr>S!..MaxNumberSchreierGens if IsIdenticalObj( table[2*i-1], table[2*i] ) then [22*]= ( table[*-1]) fi; od; # standardize the table StandardizeTable( table ); return success result r.closed := true; return true; end; GENSS_ToddCoxeterExpandLimit := function(r,newlimit) local app,g,l,limit,next,prev,table; #else app := r# (ewpres,lsgs[]1]+)l[gs[[]+sb]* table := app[1]# GENSS_Prodli,k[]sb)java.lang.StringIndexOutOfBoundsException: Rang next := app[2]; prev := app[3]; limit := r.limit; next[newlimit] := 0; prev[newlimit] := newlimit-1; for g in table do g[java.lang.StringIndexOutOfBoundsException: Index 30 out of bounds for l in [ limit+2 .. newlimit-1 ] do next[l] := l+1; prev[l] := l-1; for g in table do g[ od; next[limit+1] := limit+2; prev[limit+1] := 0; for g in table do g[limit+1] := 0; od; app[6] := limit+1; # this is firstFree r.limit := newlimit; app[7] := newlimit; # this is lastFree end; GENSS_TCAddRelators := function(r,newrels) local i,relsGen; :=RelatorRepresentatives(; newrels := java.lang.StringIndexOutOfBoundsException: Index 31 out of bounds for l nums, for i in [1..Length(relsGen)] do relsGen[,newrelsi)java.lang.StringIndexOutOfBoundsException: Range [36, 37) out od; end; ## VerifyStabilizerChainTC5 := ## Info(Info3,"deleted alive maximal"); ## local FindTwoWords,Grels,Hrels,Prels,allgens,cosetlimit,done,el,f, ## gens,gensi,hom,k,l,li,newpres,newrel,nrcosets,nrgens,o,opgens, # ords,res,rrels,sbstronggens,strongi,,cw # # ifS!< falsethen ## pres : :=2 = limit ## if IsList(pres) then return pres; fi; ## else # pres# nextjava.lang.StringIndexOutOfBoundsException: Range [10, 9) out of bounds ## fi; ## Info(InfoGenSS,1,"Verifying stabilizer chain in layer ",S!.layer); ## ## o := S!.orb; ## nrgens := Length(o!.gens); ## sb := S!.strongbelow; ## ## f := FreeGroup(sb+nrgens); ## gens := GeneratorsOfGroupg :ListWithIdenticalEntries( ,); ## gensi := List(gens, length, length2, # length of relator f not( ^ or - rels hen ## allgens := 0*[1..2*Length(gens)]; ## allgens{[1,3..2*Length(java.lang.StringIndexOutOfBoundsException: Index 32 out ## allgens{[2,4..2*Length(gens)]} := gensi; ## # make therows therelatorsand distribute over relsGen ## Hrels := ResultOfStraightLineProgram ## if S!.opt.Projective then ## ords := List([1..nrgens],i->ProjectiveOrder(o!.gens[i])); ## java.lang.StringIndexOutOfBoundsException: Range [12, 1) out of bounds for len ## #T this code code use - its faster ## fi; ## Prels := List([1..nrgens],i->gens[i+sb]^ords[i]); ## Grels := []; ## stronggens := StrongGenerators(S); ## strongi := List(stronggens,x->x^-1); ## opgens := 0*[1..2*Length(stronggens)]; ## opgens{[1,3..2*Length(stronggens)-1]} := stronggens; ## opgens{[2,4..2*Length(stronggens)]} : p1 : 2 * p; ## ## # Now start up a coset enumeration else ## cosetlimit := Maximum(QuoInt(7 * Length(o),6),Length(o)+5); ## Info(InfoGenSS,2,"Starting coset enumeration with limit ", ## cosetlimit," and ",Length(Hrels), ## "+",Length(Prels),"+",Length(Grels)," relations..."); ## rels := Concatenation(Hrels,Prels); ## tc := nums[j] := p1; cols[j] := tablep1]; ##done : GENSS_DoToddCoxeter(tc); # ## ## FindTwoWords ## TraceWord,,,,j,,,,ts, ## ptsrev,schgen,schpt,w1,w2,x,y; ## nrcosets := Length(table[1]); # java.lang.StringIndexOutOfBoundsException: Range [19, 18) out of bounds for len ## cosetsrevtab := 0*[1..nrcosets]; ## cosetsrevtab[1] := 1; ## pts := [o[1]]; ## ptsrev := 0*[1.. 1) ## ptsrev[1] := 1; ## schpt := [fail]; # the Schreier tree ## schgen := [fail]; ## i := 1; ## TraceWord := function(pos) ## local w; ## w := []; ## while pos > 1 do ## Add(w,schgen[pos]); ## pos := schpt[pos]; ; ## return Reversed(w); ## end; ## while i <= ## while i <= Length =app[]; ## for j in [1..Length(opgens)] do ## : ## if x <> 0 then # image is defined: ## if cosetsrevtab[x] = 0 then # not visited ## Add(cosets,x); ## ## newand forundefined # [x := java.lang.StringIndexOutOfBoundsException: Range [51, 52) out of bounds for ##schpt[new] := i; ## schgen[new] := j; ##java.lang.StringIndexOutOfBoundsException: Index 6 out of bounds for length 0 ## y := Position(o,pt); ## [] 0 ## Add(pts,pt); ## ptsrevifIsStraightLineProgElm(rel)then ## # have anewcoset by a word that ## # maps the starting point of the orbit to the ## # same point as the one of another coset! ## w1 := TraceWord(ptsrev[y]); ## w2 := TraceWord(new); ## return [w1,w2]; # fijava.lang.StringIndexOutOfBoundsException: Index 31 out of bounds for length ## fi; ## fi; ## od; ## i := i + 1; ## od; ## nnewlimit]: ; ## return fail; ## end; ##java.lang.StringIndexOutOfBoundsException: Range [11, 4) out of bounds for len ## while true do # will be left by return eventually ## if done = true then ## nrcosets := Length(tc.table[1]); ## Info(InfoGenSS,2,"Coset enumeration found ",nrcosets," cosets."); ## if nrcosets = Length(o) then ### Verificationis OK,nowbuild presentation: ## l := GeneratorsWithMemory( ## r.imit :=newlimit ## newpres := ResultOfStraightLineProgram(pres, ## l{[1 ## for k in [1..nrgens] do ## Add(newpres,l[k+sb]^ords[k]); ## od; ## hom := GroupHomomorphismByImagesNC(f,Group(l),gens,l); ## for k in Grels do ## Add(newpres,ImageElm(hom,k)); #java.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0 ## Info(InfoGenSS,2,"Found presentation for layer ",S!.layer, ## # FindTwoWordsG,rels,Prels,llgens,cosetlimitdone,lfjava.lang.StringIndexOutOfBo ## return SLPOfElms(newpres); ## elif nrcosets < Length(o) then ## Error("This cannot possibly have happened!"); ## return [fail,S!.layer]; ## else # nrcosets > Length(o) ## Info(InfoGenSS,2,"Too many cosets, we must have forgotten ", ## "another relation!"); ## Error("This cannot possibly have happened2!"); ## return [fail,S!.layer]; ## fi; ## fi; ## # Now we have to find another relation, we do a breadth-java.lang.StringIndexOutOfBoundsE ## # search through the already defined cosets to find two cosets # that different but to be because the ###subgens : {[.S!.strongbelow}java.lang.StringIndexOutOfBoundsException: Index 45 ## words := FindTwoWords(o,opgens,tc.table); ## el := EvaluateWord## : List([.rgens]i>rojectiveOrder(o!g[i); ## r :# ## if not(r.isone) then ## # Error, we found a new stabilizer element! ## return [fail,S!.layer,el]; ## fi; ## newrelList(x->-); ## /EvaluateWord(allgens,words[2])) ## / java.lang.StringIndexOutOfBoundsException: Index 45 out of bounds for length ## GENSS_TCAddRelatorswhile firstDef<>0 ## Add(Grels,newrel[1]); ## if Length(words[1]) > 0 then ## tc.[10] := words[1[1]; ###"+ :table] ## # More difficult: ##words =ExtRepOfObj([]; ## if words[2] > 0 then ## tc.app[10]inv: :=function,t) # ## tc.app[10] := 2*words[1]; ##fi ## fi; # Print("<\capp8]: firstDef; ## #for i vtab[] : ; ## # for j in [1..Length(allgens)]r.rdef : nrdef; ## r.nrmax = nrmaxjava.lang.StringIndexOutOfBoundsException: Range [37, 38) out of # tc : java.lang.StringIndexOutOfBoundsException: Range [37, 38) out of bounds for ## # tc.app[10] := j; ## # tc.nrdel := tc.nrdel + MakeConsequences( tc.app ); ## # fi; # # od; ## #od; ## tc.app[11] := 1; ## tc.nrdel := tc.nrdel + MakeConsequences( tc.app ); ## #Print("-\c"); ## done := GENSS_DoToddCoxeter(tc); ## #Print(">\c"); ## if Length(Grels) mod 100 = 0 then ## #Print("\n"); ## Info(InfoGenSS,2,"Currently using ",Length(Hrels# correspondingpointsareequal: ## Length(## Length(Prels,c.; # fi; ## od; ## end; ## ## VerifyStabilizerChainMax := ## function( S ) ## local bl,count,d,gens,i,java.lang.StringIndexOutOfBoundsException: Index 36 out ## ## ## / (rslp,ens); # S.tab < false ## res :=ls,Append(relsGeni,wrels[]; ## java.lang.StringIndexOutOfBoundsException: Range [15, 14) out of bounds for le ## fi; ## Info(InfoGenSS,1,"Verifying java.lang.StringIndexOutOfBoundsException: Range [0 ## " o ##function(S) ## ## count := 0; # gens: ShallowCopyS(S!.rb#if [2]>0 then # invtab #pres,rels,s,tc,java.lang.StringIndexOutOfBoundsException: Range [70, 69) ## k := Length(gens); ## for i in [1..k] do ## x := gens[i]; ## xi := x^-1; ## p := Position(gens,xi); ## if p = fail then ## Add(gens,xi); ## invtab[i] := Length(gens); ## invtab[Length(gens)] := i; ## else ## invtab[i] := p; ## fi; ## od; ## o := Orb(gens,S# f:= FreeGroup(+nrgens); ## Enumerate(o); ## # Length()< LengthS!orb)then ## Error("something#allgens{1,..Length# ("-") # java.lang.StringIndexOutOfBoundsException: Range [19, 18) out of bounds for len ## fi; ## #have fjava.lang.StringIndexOutOfBoundsException: Range [43, 42) out of bounds fo ## # are again # java.lang.StringIndexOutOfBoundsException: Range [21, 20) out of bou ## ("",ngth(),"relations."); ## for x in gens do ## if o!.op(o[1],x) = o[1] then ## count := count +#f; ## if S!.stab = false then ## r := rec( isone := isone(x) ); # stronggens:SS); ## r := SiftGroupElement(S!.stab,x); ## # isone = .sOne; ## if not(r.isone) then # return [fail,S!.layer,x,"generator"]; ## fi; ## fi; ## od; ## ## bl :# Info(InfoGenSS,1,Verifyingstabilizer inl# cosetlimit" Length(java.lang.String f in 1..o. java.lang.StringIndexOutOfBoundsException: Index 33 out of bounds for le ## Info(InfoGenSS,2,"Testing in depth ",d," (of ",o!.java.lang.StringIndexOutOfBoundsE ## o!.depthmarks[d+1]-o!.depthmarks[d]," points (## FindTwoWords:= (,,table ## for i in#k =Length(; ## # These indices contain points in depth d ## if bl[o!.schreierpos[i]] then ## # this subtree does not need to be done! ## bl[i] := true; # mark java.lang.StringIndexOutOfBoundsException: Range [0, 51) out ## else ## x := o[i]; ## for j in [1..Length(gens)] do ## if j <> invtab[o!.schreiergen[i]] then ## y := o!.op(x,gens[j]); ## p := Position(o,y# =fposjava.lang.StringIndexOutOfBoundsException: Index 38 out o ## # if p > i then this is a new point, do nothing # p< hen ## count := count + , ## w1 := TraceSchreierTreeForward(o,i); ## w2 := TraceSchreierTreeForward(o,p); ## :=(EvaluateWord(gens,1)gens[j]) ## /EvaluateWord(gens,w2); ## #Print(Length(gens),w1,j,w2,"\n"); ## if## tc.nrdel : .nrdel MakeConsequences(tc.app ; =S!stab; ## r := rec( isone := isone(sodjava.lang.StringIndexOutOfBoundsException: Index 7 out ## else ## r := SiftGroupElement(S!.stab,s); ## fi; ## if not(r.isone) then ## return [fail,S!.layer,s,"schreier gen"]; ## fi; ##fi; ## if p < o!.depthmarks[d] then ## # this goes up in # done =GENSS_DoToddCoxeter ## # Schreier generator is OK, we do not have to ## # look below! ## bl[i] := true; ## break; # fi; ## fi; ## od; ## fi; ## od; ## od; ## Info(InfoGenSS,2,"Have sifted ",count," elements."); ## return true; ## end; ############################################################################# # The following operations apply to stabilizer chains: ################################################ InstallMethod( Size, "for a stabilizer chain", [ IsStabilizerChain and IsStabilizerChainByOrb ], function( S ) local size; size := 1; while S <> false do ize: * (!.orb; S := S!.stab; od; return size; end ); InstallMethod( \in, "for a group element# : i]; [ IsObject, IsStabilizerChain and IsStabilizerChainByOrb ], function( el, S ) local r; r := SiftGroupElement(S,el); if r.isone then return true; else return false; fi; end ); GENSS_VIEWDEPTH := 0; # to please the interpreter (java.lang.StringIndexOutOfBoundsException: Index 17 out of bounds for length 8 [ IsStabilizerChain and java.lang.StringIndexOutOfBoundsException: Index 45 out of function( S ) local i; if not(IsBound(GENSS_VIEWDEPTH)) then GENSS_VIEWDEPTH := 0; Print("\n"; fi; for i in [1..GENSS_VIEWDEPTH] do Print(" "); od; Print("<stabchain size=",Size(S)," orblenGENSS_VIEWDEPTH : 1; # ; then Print("\n"); ViewObj(S!.stab); fi; GENSS_VIEWDEPTH := java.lang.StringIndexOutOfBoundsException: Index 27 out of boun if GENSS_VIEWDEPTH < 0 then Unbind(GENSS_VIEWDEPTH); fi; end ); Unbind(java.lang.StringIndexOutOfBoundsException: Range [63, 22) out of bounds f InstallMethod( BaseStabilizerChain, "generic method for stabilizer chains", [IsStabilizerChain], function( S ) ,tsojava.lang.StringIndexOutOfBoundsException: Index 21 out of bounds for length pts := []; ops := []; SS := S; while SS <> false do Add(pts,SS!.orb[1]); Add(ops,SS!.orb!.op); SS := SS!.stab; od; return rec( points := pts, ops := ops ); end ); InstallMethod( SiftBaseImage, "generic method for genss stabilizer chains", [IsStabilizerChain, IsList], function(S,bi) local l, SS, i, o, pos; l := Length(bi); SS := S; i := 1; while i <= Length(bi) do o := SS!.orb; pos := Position(o,bi[i]); if pos = fail then return false; fi; while pos > 1 do if o!.memorygens then bi{[i..l]} GENSS_VIEWDEPTH:= 0 # interpreter o!.gensi[o!.schreiergen[pos]]!.el,SS); else bi{[i..l]} := GENSS_MapBaseImage(bi{[i..l]}, o!.gensi[o!.schreiergen[pos]],SS); fi; pos := o!.schreierpos[pos]; od; if o[1] <> bi[i] then Error("this should not have happened, tell the authors"); fi; i := i + 1; SS := SS!.stab; od; eturntrue; end ); InstallMethod( IsProved, "for a stabilizer chain", [ IsStabilizerChain and IsStabilizerChainByOrb ], function( S ) return S!.proof; end ); java.lang.StringIndexOutOfBoundsException: Range [53, 13) out of bounds for leng [ IsPermGroup, IsStabilizerChain and IsStabilizerChainByOrb ], function( g, S ) base; if= Error(java.lang.StringIndexOutOfBoundsException: Range [17, 16) out of bounds fo java.lang.StringIndexOutOfBoundsException: Range [15, 14) out of bounds for leng else SetSize(g,Size(S)); fi; base := BaseStabilizerChain(S); if not(ForAll(base.points,IsInt) and ForAll(base.ops,x->x=java.lang.StringIndexOutOf return StabChainOp(g); fi; java.lang.StringIndexOutOfBoundsException: Range [11, 10) out of bounds for leng size := Size(S) )); end ); InstallGlobalFunction( GENSS_FindGensStabilizer, function( S, pt, opt ) # Assumes that S is a correct stabilizer chain for the group generated by # its gens (in S!.orb!.gens) and that pt lies in the first orbit S!.orb. # Finds an SLP to express generators of the stabilizerod; # the group in terms of the gens. # Assumes that the group and the first stabilizer in the chain are # non-trivial. local gens,g,pos,mgens,word,cosetrep,invgens,pr,stabgens,size,stabsize, randel,newpt,ptpos,ptcosetrep,el,SS,stab,i; gens := S!.orb!.gens; g := GroupWithGenerators(gens); SetSize(g,Size(S)); pos := Position(S!.orb,pt); if pos = fail then Error("point pt must lie in first orbit"); return fail; fi; if localbase; if not(IsBound(opt.scramblefactor)) then opt.scramblefactors := 0; fi; if not(IsBound(opt.maxdepth)) then opt.maxdepth := 200; fi; if not(IsBound(opt.addslots)) then opt.addslots := Maximum(0,5-Length(gens)); fi; # Give the gens a (new) memory: if IsObjWithMemory(gens[1]) then ase =BaseStabilizerChainS; else mgens := GeneratorsWithMemory(gens); fi; # FIXME: use GENSS_Prod if pos = 1 then :=mgens[1^0; else word := TraceSchreierTreeForward(S!.orb,pos); cosetrep := Product(mgens{word}); java.lang.StringIndexOutOfBoundsException: Range [13, 11) out of bounds for leng # Set up the product replacer: pr := ProductReplacer( mgens, opt ); stabgens := []; # here we collect the stabilizer generators size := 1; := (S.; #this exist! repeat randel := Next(pr); newpt := S!.orb!.op(pt,randel!.el); ptpos := Position(S!.orb,newpt); java.lang.StringIndexOutOfBoundsException: Range [40, 12) out of bounds for leng if java.lang.StringIndexOutOfBoundsException: Index 17 out of bounds for length 1 ptcosetrep := Product(invgens{word}); el := randel * ptcosetrep; else el := randel; fi; if pos <> 1 then java.lang.StringIndexOutOfBoundsException: Range [31, 14) out of bounds for leng fi; # now el is an element in the stabilizer of pt if S!.pos := Position(SS := Stajava.lang.StringIndexOutOfBoundsException: Range [34, Info(InfoGenSS,3,"Found trivial stabilizer element."); else Add(stabgens,el); stab := GroupWithGenerators(stabgens); SS := StabilizerChain(stab,rec( Base := S, IsOne := S!.IsOne )); Info(InfoGenSS,1,"Have group size ",Size(SS)," (of ", stabsize,")"); if ( =size java.lang.StringIndexOutOfBoundsException: Range [23, 22) out of bounds for leng else =Size(; fi; fi; until #the group terms the gens. # Assumesthat groupand thefirst thechainare i := 1; repeat Info(InfoGenSS,1,"Need ",Length(stabgens)+1-i," generators."); i := i + 1; if i < Length(stabgens) then stab := Group(stabgens{[i..Length(stabgens)]}); = stab,ec(Base :=S = S!.); size := Size(SS); else size gSize(); fi; until size < stabsize; return SLPOfElms(stabgens{[i-1..Length(stabgens)]}); end ); InstallGlobalFunction( GENSS_FindShortGensStabilizerOld, function( S, pt ) # Assumes that S is a correct stabilizer chain for the group generated by # its gens (in S!.orb!.gens) and that pt lies in the first orbit S!.orb. # Finds an SLP to express generators of the stabilizer of pt in # the group in terms of the gens. # Assumes that the group and the first stabilizer in the chain are # non-trivial. local gens,g,pos,mgens,word,cosetrep,invgensword := TraceSchreierTreeForwardS.rbpos ,,,tcosetrep,elSS,tab,,jjava.lang.StringIndexOutOfBoundsException: Index 55 out :=S!orb!gens g := GroupWithGenerators(gens); SetSize(g,Size(S)); pos := Position(S!.orb,pt); if pos = fail then Error("point pt must lie in first orbit"); return fail; fi; # Give the=Size(S!.stab; # hasto ! IsObjWithMemorygens[1]) mgens = (Listgens,-x!.el)); else mgens := GeneratorsWithMemory(gens); fi; if pos = 1 then cosetrep := mgens[1]^0; else word := TraceSchreierTreeForward(S!.orb,pos); cosetrep := Product(mgens{word}); invgens := if j > Lengthmgens) # Set up the search: pr := ShallowCopy(mgens); i := 1; # counts points j := 1; # counts gens stabgens := []; # here we collect the stabilizer generators size := 1; stabsize := Size(S!.stab); # this has to exist! repeat Add(pr,pr[i]*mgens[j]); j := j + 1; if j > Length(mgens) then j := 1; i := if pos <> 1 fi; java.lang.StringIndexOutOfBoundsException: Index 11 out of bounds for length 11 newpt := S!.orb!.op(pt,randel!.el); ptpos := Position(S!.orb,newpt); word := TraceSchreierTreeBack(S!.orb,ptpos); if Length(word) > 0 then ptcosetrep := java.lang.StringIndexOutOfBoundsException: Index 31 out of bounds fo el := randel * ptcosetrep; else el := randel; ; <> stabsize")) el := el * cosetrep; fi; # now el is an element in the stabilizer of pt if S!.IsOne(el) then Info(InfoGenSS,2,"Found trivial stabilizer element."); else Add(stabgens,el); stab := GroupWithGenerators(stabgens); := StabilizerChain(rec(Base =S IsOne = S.)); Info(InfoGenSS,1ptpos := Position(orb,newpt); stabsize,")"); if Size(SS) = size then Remove(stabgens); else size := Size(SS); java.lang.StringIndexOutOfBoundsException: Range [15, 16) out of bounds for leng fi; el =randel * ; # Try leaving out java.lang.StringIndexOutOfBoundsException: Index 12 out of bounds i := 1; repeat Info(InfoGenSS,1,"Need ",Length(stabgens)+1-i," generators."); i := i + 1; if i < Length(stabgens) then stab := Group(stabgens{[i..Length(stabgens)]}); SS := StabilizerChain(stab,rec(Base := S, IsOne := S!.IsOne)); size := Size(SS); else size := 0; fi; size stabsizejava.lang.StringIndexOutOfBoundsException: Range [26, 27) out of bou return SLPOfElms(stabgens{[i-1..Length(stabgens)]}); end ); InstallGlobalFunction( GENSS_FindShortGensStabilizer, function( gens, pt, op, grpsize, stabsize, S ) # Assumes that S is a correct stabilizer chain a supergroup of the # group g generated by gens. pt and op is an action for g. # grpsize is the Size of g and Error("Something is wrong, stabsizewas small"); # Finds an SLP to express generators of the stabilizer of pt in # g in terms of the gens. # Assumes that g and the orbit of pt under g are non-trivial. local SS,el,g,i,invgens,j,mgens,newpt,orb,pr,ptcosetrep,ptpos,randel,size, stab,stabgens,word; g := GroupWithGenerators(gens); InfoInfoGenSS,2,Enumerating orbitof ",grpsize/tabsize) orb := Orb(gens,pt,op,rec(if (SS < then Enumerate(orb); # Give the gens a (new) memory: if IsObjWithMemory(gens[1]) then mgens := GeneratorsWithMemory(List(gens,x->x!.el)); else mgens := GeneratorsWithMemory(gens); fi; invgens := List(mgens,x->x^-1); # Set up the search: pr := ShallowCopy(mgens); i := 1; # counts points j := 1; # counts gens stabgens := []; # here we collect the stabilizer generators size := 1; repeat Add(pr, # the stabilizers in the chain, each in terms of the generators of the j := j + 1; if j > Length(mgens) then j := 1; i := i + 1; fi; del:=pr[Length(pr)] newpt := op(pt,randel!.el); ptpos := Position(orb,newpt); word:=TraceSchreierTreeBack(orb,ptpos); if Length(word) > 0 then ptcosetrep := Product(invgens{word}); el := randel * ptcosetrep; else el := randel; fi; # now el is an element in the java.lang.StringIndexOutOfBoundsException: Index 46 out of !.sOne( then Info(InfoGenSS,3,"Found ], else Add(stabgenslocal Sr stabS: (g; SS := StabilizerChain(stab,rec( Base := S, IsOne := S!.IsOne )); Info(InfoGenSS,1,"Have group size ",Size(SS)," (of ", stabsize,")"); if Size(SS) = size then RemoveInstallMethod(\n f a storedstabilizer , else size := Size(SS); fi; fi; until size >= stabsize; if size > stabsize then Error("Something is wrong, stabsize limit was too small"); return fail; java.lang.StringIndexOutOfBoundsException: Index 7 out of bounds for length 7 out the first generators found: i := 1; Info(InfoGenSS,1,"Need ",Length(stabgens)," generators."); while i < Length(stabgens) do stab := Group(stabgens{Concatenation([1..i-1],[i+1..Length(stabgens)])}); SS := StabilizerChain(stab,rec(Base := S, IsOne := S!.java.lang.StringIndexOutOfBoundsE if Size(SS) < stabsize then i := i + 1; # keep generator else Remove(stabgens,i); Info(InfoGenSS,1,"Need ",Length(stabgens)," generators."); fi; od; return SLPOfElms(stabgens); end ); InstallGlobalFunction( SLPChainStabilizerChain, function( S, gens ) #S stabilizer chain assumed to be correct for the group g generated # by generators gens. # Returns a list of straight line programs expressing successively # the stabilizers in the chain, each in terms of the generators of the # previous, the first in terms of gens. local l,ll,slp; l := []; ll := [Size(S)]; while S!.stab <> false do Info(InfoGenSS,1,"Working on group with size ",Size(S),"..."); slp := GENSS_FindShortGensStabilizer( gens, S!.orb[1], S!.orb!.op, Size(S), Size(S!.stab), S ); Add(l,slp); Add(ll,Size(S!.stab)); gens :=slp,gens (nfoGenSS1" , Length(LinesOfStraightLineProgram(slp))," lines toend ); Length(gens)," generators."); S := S!.stab; od; return rec(slps := l,sizes := ll); end ); InstallGlobalFunction( GENSS_MakeIterRecord, function( S ) local SS,Slist,it; Slist := [S]; SS := S!.stab; while SS <> false do Add(Slist,SS); = SS!.tab; od; it := rec( NextIterator := GENSS_GroupNextIterator, IsDoneIterator := GENSS_GroupIsDoneIterator, ShallowCopy := GENSS_GroupShallowCopy, pos := ListWithIdenticalEntries( Length( S!.base ), 1 ), Slist := Slist, cache := ListWithIdenticalEntries( Length(S!.base), One(S!.orb!.gens[1]) ), ne :One(!orb![], lens := List(Slist,x->Length(x!.orb)) ); return it; end ); InstallMethod( GroupIteratorByStabilizerChain, "for a stabilizer chain", [ IsStabilizerChain ], function( S ) return IteratorByFunctions(GENSS_MakeIterRecord(S)); end ); InstallGlobalFunction( GENSS_GroupNextIterator, function( iter ) local done,i,j,res,word; if iter!.pos[1] > iter!.lens[1] then return fail; fi; res := iter!.cache[Length(iter!.cache)]; # Now step forwards: i := Length(iter!.Slist); repeat iter!.pos[i] := iter!.pos[i] + 1; if iter!.pos[i] > iter!.lens[i] then iter!.pos[i] := 1; i := i - 1; done := false; else done := true; fi; until done or i = 0; if i = 0 then iter!.pos[1] := iter!.lens[1]+1; return res; fi; # we are done word := TraceSchreierTreeForward(iter!.Slist[i]!.orb,iter!.pos[i]); iter!.cache[i] := Product(iter!.Slist[i]!.orb!.gens{word}); if i > 1 then iter!.cache[i] := iter!.cache[i] * iter!.cache[i-1]; fi; for j in [i+1..Length(iter!.Slist)] do iter!.cache[j] := iter!.cache[j-1]; od; return res; end ); InstallGlobalFunction( GENSS_GroupIsDoneIterator, function( ) returnIsGroup,IsObject], end ); InstallGlobalFunction( GENSS_GroupShallowCopy, function( iter ) java.lang.StringIndexOutOfBoundsException: Index 8 out of bounds for length 7 end ); ############################################################################# # We can store a stabilizer chain in the attribute # StoredStabilizerChain, then "ENSS method arbitrary , ############################l ; InstallMethod( SetStabilizerChain, "for a group and a stabilizer chain", [IsGroup, IsStabilizerChain], function(g,S) if IsIdenticalObj(S!.IsOne,IsOne) and HasSize(g) and Size(g) <> Size(S) then Error("you try to set a stabilizer chain for the wrong group"); return; fi; (,) end ); InstallMethod( Size, "for a group with a stored stabilizer chain", [IsGroup and HasStoredStabilizerChain], function(g) return Size(StoredStabilizerChain(g)); end ); InstallMethod( Size, "for a perm. group with a stored stabilizer chain", [IsPermGroup and HasStoredStabilizerChain], function(g) return Size(toredStabilizerChain()); end ); InstallMethod( Size, "for a group with a stored stabilizer chain", [IsGroup and HasStoredStabilizerChain and IsHandledByNiceMonomorphism], function( return InstallMethod(ORB_IsElementInStabilizerChain, "GENSSmethod arbitrary " InstallMethod( \in, "for a group elm and a group with stored stabilizer chainfunction(el,S) [IsPerm, IsPermGroup and HasStoredStabilizerChain], 1, function(x, g) local S,r; S := StoredStabilizerChain(g); =SiftGroupElement(,x; return r.isone; gm: GroupWithMemory(); InstallMethod( \in, "for a group elm and a group with stored stabilizer chain", [IsObject, IsMatrixGroup and HasStoredStabilizerChain and IsHandledByNiceMonomorphism], function(x, g) local S,r S IsList,IsObject,IsFunction, r := SiftGroupElement(Sx) return r.isone; ) InstallMethod( \in, "for a group java.lang.StringIndexOutOfBoundsException: Index 36 [IsObject, IsGroup and [IsList, IsObject, IsFunction,java.lang.StringIndexOutOfBounds function[IsList, function(l xop) local S,r; S := StoredStabilizerChain(g); r := SiftGroupElement(S,x); return r.isone; end ); InstallOtherMethodWithRandomSource( Random, "for a random source and a stabilizer chain", [ IsRandomSource, IsStabilizerChainByOrb ], functionrs,S) local x,pos,w; x := One(S!.orb!.gens[1]); while S <> false do pat,pos,r,s,,,stabgens,stabsizeest,w1w2InstallMethod Stab,"yOrb java.lang.StringIn w := TraceSchreierTreeForward(S!.orb,pos); x := GENSS_Prod( local S,el,java.lang.StringIndexOutOfBoundsException: Range [31, 30 S := S!.stab; od; ; ); InstallMethodWithRandomSource( Random, "for a random source and a group with a stored stabil [ IsRandomSource, IsGroup and java.lang.StringIndexOutOfBoundsException: Index 44 ou functionrs,g (,StoredStabilizerChain(, roof : true ; InstallMethodWithRandomSource( java.lang.StringIndexOutOfBoundsException: Index 3 [ IsRandomSource, IsPermGroup and HasStoredStabilizerChain ], 10, function(rs, g) return Random(rs, StoredStabilizerChain(g)); end ); InstallMethodWithRandomSource( proof := true ) [ IsRandomSource, IsMatrixGroup and IsHandledByNiceMonomorphism and HasStoredStabilizerChain ], function(rs, g) return Random(rs, toredStabilizerChain(g) end); InstallMethod( SizeMC, "for a group and an error bound", [IsGroup, IsRat], function( G, err ) local S; S := StabilizerChain(G,rec( ErrorBound := err )); return Size(S); end ); InstallMethod( SizeMC, "for a permutation group and an error bound, genss", [IsGroup and IsPermGroup, IsRat],ifcount < 10 then function( G, err ) local S; S := StabilizerChain(G,rec( ErrorBound := err )); return Size(S); end ); InstallGlobalFunction( GENSS_ImageElm, function( data, x ) local r; r := SiftGroupElementSLP(data!.Sg,x); if not(r.isone) then return fail; fi; return ResultOfStraightLineProgram(r.slp,data!.stronggims); end ); i ( then function( data, x ) local r; r := SiftGroupElementSLP(data!.Si,x); if not(r.isone) then return fail; fi; return ResultOfStraightLineProgram(r.slp,data!.strongipre); end ); InstallGlobalFunction( GroupHomomorphismByImagesNCStabilizerChain, function( g, h, images, opt1, opt2 ) local Sg,Si,data,gm,im,slpstrongg,slpstrongi,strongg,stronggims, strongi,strongipre; gm := GroupWithMemory(g); Sg := StabilizerChain(gm,opt1); strongg := StrongGenerators(Sg); slpstrongg = SLPOfElmss); ForgetMemory(Sg); stronggims := ResultOfStraightLineProgram(slpstrongg,images); im := GroupWithMemory(images) =errorprob 2; Si := StabilizerChain(im,opt2); strongi := StrongGenerators(Si); slpstrongi := SLPOfElms(strongi); ForgetMemory(Si); strongipre := ResultOfStraightLineProgram(slpstrongi,GeneratorsOfGroup(g)); data := rec( Sg := Sg, stronggims := stronggims, Si := Si, strongipre := strongipre, slpstrongg := slpstrongg, slpstrongi := slpstrongi ); return GroupHomByFuncWithData( g, h, GENSS_ImageElm, false, GENSS_PreImagesRepresentative, data ); end ); InstallMethod( ORB_StabilizerChainKnownSize, "GENSS method for arbitrary groups", [IsGroup,IsPosInt], function(g,size) lizerChaingand size = Size(java.lang.StringIndexOutOfBoundsException: Index 37 out of bounds for return StoredStabilizerChain(g); fi stabilizerchain := stabchain, return StabilizerChain(g,rec( Size := size )); end ); InstallMethod( ORB_BaseStabilizerChain, "GENSS method for if java.lang.StringIndexOutOfBoundsException: Range [19, 18) out o [IsStabilizerChain], BaseStabilizerChain ); InstallMethod( ORB_StabilizerChainKnownBase, "method forarbitrarygroups"java.lang.StringIndexOutOfBoundsException: Index 38 ou [IsGroup,IsObject], g,) iffi; tabilizerChaing); fi; if base = fail then returnif IsOne( then else return StabilizerChain( g, rec( Cand := ShallowCopy(base) ) ); fi; end ); InstallMethod( ORB_SizeStabilizerChain, "GENSS method for arbitrary groups", [IsStabilizerChain], Size ); InstallMethod( ORB_IsWordInStabilizerChain, "java.lang.StringIndexOutOfBoundsException: Range [9, 8) out of bounds for lengt [IsList,IsListreturn StabMC(GroupWithGenerators(l),x,op,opt); function(word, gens, gensi, S) local x, b, if Size(SInstallMethod( " Stab " x := [IsGroup, IsObject, IsFunction, IsRecord], return S!.IsOne(x); i; b := BaseStabilizerChain(S); ops := b.ops; b := b.points; for i in [1..Length(word)] do if word[i] > 0 then b := List([1..Length(b)],j->ops[j](b[j],gens[word[i]])); else b := List([1..Length(b)],j->ops[j](b[j],gensi[-java.lang.StringIndexOutOfBoundsExce fi; od; return SiftBaseImage(S,b); end ); InstallMethod ORB_IsElementInStabilizerChain "GENSS method for arbitrary groups", [IsObject, IsStabilizerChain], function(el, S) return# forwhich xg)gnot ecessarily U! end )# pruner either a taking ifHasSizeg nd ############################################################################# # The following methods are cache := WeakPointerObj([[Sorb!.gens1^,]]]; #########S(,g)java.lang.StringIndexOutOfBoundsException: Index 46 out of bounds (Stab,"dd empty record"java.lang.StringIndexOutOfBoundsException: Index 48 out of [ java.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0 [sGroup IsFunction LGbe agroup and the java.lang.StringIndexOutOfBoundsException: Rang return Stab(g,x,op,rec()); end ); InstallMethod( Stab, "add empty options record", [IsList, IsObject, IsFunction], l ,op) return Stab(l,x,op,rec()); end ); InstallMethod( Stab, "create group from list of generatorsreturn (,,,opt; [IsList, IsObject, IsFunction, IsRecord], function( l, x, op, opt ) return Stab(GroupWithGenerators end ); # Now the real one: InstallMethod( Stab, "by Orb orbit enumeration", [IsGroup, IsObject, IsFunction, IsRecord], ,opt java.lang.StringIndexOutOfBoundsException: Range [27, 28) out of bounds for local S,count,el,errorprob,found,gens,i,j,limit,memperpt,nrrand,java.lang.StringI pos,,,,,stabsizeestw1,w2,; GENSS_CopyDefaultOptions(GENSS,opt); if HasStoredStabilizerChain(g) then S := StoredStabilizerChain(g); if IsIdenticalObj(S!.orb!.op,op) and x in S!.orb then if S!.stab = false then =rec( : TrivialSubgroupg,size 1,proof:= y.stabilizerchain := StabilizerChain(y.stab); return y; fi; pos := Position(S!.orb,x); w1 := TraceSchreierTreeForward(S!.orb,pos); y := GENSS_Prod(S!.orb!.gens,w1); stab := Group(List(S!.stab!.orb!.gens,a->y^-1*a*y)); return rec( stab := stab, size := Size(S!.stab), stabilizerchain := StabilizerChain(stab,rec( Base := S)), proof := true ); fi; fi; # Now we have to do it ourselves: if not(IsBound(opt.ErrorBound)) then # we are working deterministically if not(HasSize(g)) then # We are basically stuffed, unless we want to check all # Schreier generators of an orbit! Error("I do not want to check all Schreier generators, ", "need size of group or ErrorBound option! lse return fail; fi; else if opt.ErrorBound < opt.StabAssumeCompleteLimit then opt.StabAssumeCompleteLimit := opt.ErrorBound; fi; fi; #we know groupsizeor Monte Carlo: errorprob := 1; limit := opt.StabInitialLimit; pat := opt.StabInitialPatience; o := Orb(g,x,op,rec( report := opt.Report, treehashsize := opt.InitialHashSize, schreier := true ) ); stabgens := []; stabsizeest := 1; pr := ProductReplacer(GeneratorsOfGroup(g), rec( scramble := opt.StabScramble, scramblefactor := opt.StabScrambleFactor, addslots := opt.StabAddSlots, maxdepth := opt.StabMaxDepth )); gens := GeneratorsOfGroup(g); # Now go through a cycle of orbit enumeration and stabilizer generation: repeat if not(IsClosed(o)) then if HasSize(g) and limit > QuoInt(Size(g),stabsizeest*2InstallGlobalFunctionComputeSubo limit := QuoInt(Size(g),stabsizeest*2)+2; fi; Info(InfoGenSS,2,"Enumerating orbit with limit ",limit); Enumerate(o,limit); if IsClosed(o) then closed, is,(); fi; fi; if errorprob < opt.StabAssumeCompleteLimit then if HasSize(g) and 2 * Length(o) * stabsizeest > Size(g) then # Done! #stab := Subgroup(g,stabgens); Length()>0 then stab := Group(stabgens); SetParent(stab,g); SetSize(stab,stabsizeest); SetStabilizerChain(stab,stabchain); else java.lang.StringIndexOutOfBoundsException: Index 24 out of bounds for length 0 stabchain := StabilizerChain(stab); fi; return rec( stab := stab, size := stabsizeest, stabilizerchain := stabchain, proof := true ); fi; w : L(t[2]+; if not IsClosed(o) then continue; fi; fi; count := 0; found := false; repeat el := Next(pr); i := 1; java.lang.StringIndexOutOfBoundsException: Range [18, 17) out of bounds for leng y := op(o[i],el); j := Position(o,y); if j <> fail then :=true else i := i + 1; fi; od; count := count + 1; until found or count >= pat; if not(IsClosed(o)) then if not(found) then =QuoInt(limit,2; else if count < 10 then limit := limit + 100; else limit :rec(treehashsize : fi; fi; fi; pat := QuoInt(pat*5,4); biliser element: Info(InfoGenSS,3,"Found a potential stabilising element...");fi; Enumerate()java.lang.StringIndexOutOfBoundsException: Index 35 out of bounds for w2 := TraceSchreierTreeForward(o,j); fijava.lang.StringIndexOutOfBoundsException: Index 19 out of bounds for length 1 if IsOne(stabel) then =errorprob 2; Info(InfoGenSSif o = then Nosolution found else Add(stabgens,stabel); if Length(stabgens) < 2 then Info(InfoGenSS,3,"Waiting for second stabiliser element."); else # Length(stabgens) >= 2 then if Length(stabgens) = 2 then Info(InfoGenSS,2,"Estimating stabilizer..."); stabchain := StabilizerChain(Group(stabgens)); java.lang.StringIndexOutOfBoundsException: Range [34, 33) out of bounds for leng else res := AddGeneratorToStabilizerChain(stabchain,stabel); if not(res) then Remove(stabgens,Length(stabgens)); errorprob := errorprob / 2; Info(InfoGenSS,2,"Error probablity now < ", errorprob); else (nfoGenSS,, "Added generator to stabilizer..."); VerifyStabilizerChainMC(stabchain,10); errorprob := 1; fi; fi; if Size(stabchain) > stabsizeest then stabsizeest := Size(stabchain); Info(InfoGenSS,2,"New stabilizer estimate: ", stabsizeest); else Info(InfoGenSS,2,"Stabiliser estimate unchanged."); fi; if HasSize(g) and 2 * Length(o) * stabsizeest > Size(g) then # Done! if Length(stabgens) > 0 then stab := Group(stabgens); SetParent(stab,g); SetSize(stab,stabsizeest); SetStabilizerChain(stab,stabchain); else tab:= TrivialSubgroup(g; stabchain := StabilizerChain(stab); fi; return rec( stab := stab, size := stabsizeest, stabilizerchain := stabchain, proof := true ); fi; if IsBound(opt.ErrorBound) and errorprob < opt.ErrorBound then turn SiftGroupElement(S,a).isone; if Length(stabgens) > 0 then =Group(stabgens; SetParent(stab,g); else java.lang.StringIndexOutOfBoundsException: Range [33, 32) out of bounds for leng stabchain := StabilizerChain(stab); fi; return rec( stab := stab, size := stabsizeest, stabilizerchain := stabchain, proof := false ); fi; fi; fi; else # no element found Info(InfoGenSS,3,"No stabiliser element found!"); fi; until Length(o) > opt.StabOrbitLimit; return fail; end ); InstallMethod( StabMC, "add empty options record", [IsGroup, IsObject, IsFunction], function( g, x, op ) return StabMC(g,x,op,rec()); end ); InstallMethod( StabMC, "add empty options record", [IsList, IsObject, IsFunction], function( l, x, op ) return StabMC(l,x,op,rec()); end ); InstallMethod( StabMC, "create group from list of generators", [IsList, IsObject, IsFunction, IsRecord], function( l, x, op, opt ) return StabMC(GroupWithGenerators(l),x,op,opt); end ); InstallMethod( StabMC, "call Stab with errorbound", [IsGroup, IsObject, IsFunction, IsRecord], function( g, x, op, opt ) if not(IsBound(opt.ErrorBound)) then opt.ErrorBound := 1/1025; fi; if not(IsBound(opt.DoEstimate)) then opt.DoEstimate := 1000; # try for 1 second fi; return Stab(g,x,op,opt); end ); InstallGlobalFunction( BacktrackSearchStabilizerChainElement, function( S, # a stabilizer chain P, # a property function G -> {true,false} g, # a group element pruner ) # Let G be a group and U being the subgroup defined by S. # Does a backtrack search in U using S for an element x of U # for which P(xg) is true. g not necessarily in U! # pruner is either false or a function taking arguments as seen below. local cache,gen,i,ii,res,t,w,x,dotree; cache := WeakPointerObj([[S!.orb!.gens[1]^0,[]]]); for i in [1..Length(S!.orb)] do ii := S!.orb!.orbind[i]; if ii > 1 then t := ElmWPObj(cache,S!.orb!.schreierpos[ii]); if t <> fail then gen := S!.orb!.schreiergen[ii]; w := EmptyPlist(Length(t[2])+1); Append(w,t[2]); Add(w,gen); t := t[1] * S!.orb!.gens[gen]; else w := TraceSchreierTreeForward(S!.orb,ii); t := Product(S!.orb!.gens{w}); fi; SetElmWPObj(cache,ii,[t,w]); x := t*g; else w := []; t := S!.orb!.gens[1]^0; x := g; fi; if S!.stab = false then if P(x) then return rec( elm := x, vec := [i], word := S!.layergens{w} ); fi; else if pruner = false or pruner(S,ii,x,t,w) then res := BacktrackSearchStabilizerChainElement(S!.stab,P,x, pruner); if res <> fail then Add(res.vec,ii,1); res.word := Concatenation(res.word,S!.layergens{w}); return res; fi; fi; fi; od; return fail; end ); InstallGlobalFunction( ComputeSuborbitsForStabilizerChain, function( S ) local SS,gens; SS := S; while SS!.stab <> false do gens := StrongGenerators(S){SS!.stab!.layergens}; SS!.suborbs := FindSuborbits(S!.orb,gens); SS := SS!.stab; od; end ); InstallGlobalFunction( BacktrackSearchStabilizerChainSubgroup, function(S,P,pruner) # Let G be a group and U being the subgroup defined by S. # Does a backtrack search in U using S for the subgroup H of U # with H := {h \in U | P(h)=true}. P must be such that H is # a subgroup of U. # pruner is either false or a function taking arguments as seen below. local SS,SSS,cache,dotree,gen,i,ii,newgens,o,res,t,w,T; cache := WeakPointerObj([[S!.orb!.gens[1]^0,[]]]); if S!.stab = false then # lowest level newgens := []; o := false; for i in [2..Length(S!.orb)] do ii := S!.orb!.orbind[i]; if o = false or not(S!.orb[ii] in o) then if ii > 1 then t := ElmWPObj(cache,S!.orb!.schreierpos[ii]); if t <> fail then gen := S!.orb!.schreiergen[ii]; w := EmptyPlist(Length(t[2])+1); Append(w,t[2]); Add(w,gen); t := t[1] * S!.orb!.gens[gen]; else w := TraceSchreierTreeForward(S!.orb,ii); t := Product(S!.orb!.gens{w}); fi; SetElmWPObj(cache,ii,[t,w]); else w := []; t := S!.orb!.gens[1]^0; fi; if P(t) then Add(newgens,t); if Length(newgens) = 1 then o := Orb(ShallowCopy(newgens),S!.orb[1],S!.orb!.op, rec( treehashsize := 200, schreier := true, log := true )); else AddGeneratorsToOrbit(o,[t]); fi; Enumerate(o); fi; fi; od; if o = false then # No solution found newgens := [S!.orb!.gens[1]^0]; o := Orb(newgens,S!.orb[1],S!.orb!.op, rec(schreier := true, log := true)); Enumerate(o); fi; SSS := rec( stab := false, size := Length(o), base := [o[1]], opt := ShallowCopy(S!.opt), layer := S!.layer, orb := o, stronggens := newgens, layergens := [1..Length(newgens)], randpool := [], IsOne := S!.opt.IsOne, proof := true, parentS := false); Objectify( StabChainByOrbType, SSS ); SSS!.opt.pr := ProductReplacer(SSS!.stronggens); SSS!.topS := SSS; # this will be changed further up! o!.stabilizerchain := SSS; o!.gensi := List(o!.gens,x->x^-1); Info(InfoGenSS,2,"Layer ",SSS!.layer," completed, size ",Size(SSS)); return SSS; else # S!.stab <> false # First go down in tree, essentially trying coset rep 1 first: SS := BacktrackSearchStabilizerChainSubgroup(S!.stab,P,pruner); newgens := []; o := false; for i in [2..Length(S!.orb)] do ii := S!.orb!.orbind[i]; if o = false or not(S!.orb[ii] in o) then if ii > 1 then t := ElmWPObj(cache,S!.orb!.schreierpos[ii]); if t <> fail then gen := S!.orb!.schreiergen[ii]; w := EmptyPlist(Length(t[2])+1); Append(w,t[2]); Add(w,gen); t := t[1] * S!.orb!.gens[gen]; else w := TraceSchreierTreeForward(S!.orb,ii); t := Product(S!.orb!.gens{w}); fi; SetElmWPObj(cache,ii,[t,w]); else w := []; t := S!.orb!.gens[1]^0; fi; if pruner = false or pruner(S,ii,t,t,w) then res := BacktrackSearchStabilizerChainElement(S!.stab,P,t, pruner); if res <> fail then Add(newgens,res.elm); if Length(newgens) = 1 then o := Orb(Concatenation(StrongGenerators(SS),newgens), S!.orb[1],S!.orb!.op, rec( treehashsize := Maximum(100,QuoInt(Length(S!.orb),3)), schreier := true, log := true )); else AddGeneratorsToOrbit(o,[res.elm]); fi; Enumerate(o); fi; fi; fi; od; if o = false then # No solution found o := Orb([S!.orb!.gens[1]^0],S!.orb[1],S!.orb!.op, rec(schreier := true, log := true)); Enumerate(o); fi; SSS := rec( stab := SS, size := Length(o)*Size(SS), base := SS!.base, opt := ShallowCopy(S!.opt), layer := S!.layer, orb := o, stronggens := SS!.stronggens, layergens := [1..Length(SS!.stronggens)+Length(newgens)], randpool := [], IsOne := SS!.opt.IsOne, proof := true, parentS := false ); Add(SSS.base,o[1],1); Append(SSS.stronggens,newgens); SSS.opt.pr := ProductReplacer(ShallowCopy(SSS.stronggens)); Objectify( StabChainByOrbType, SSS ); SS!.parentS := SSS; T := SSS; while T <> false do T!.topS := SSS; T := T!.stab; od; o!.stabilizerchain := SSS; o!.gensi := List(o!.gens,x->x^-1); Info(InfoGenSS,2,"Layer ",SSS!.layer," completed, size ",Size(SSS)); return SSS; fi; end ); InstallMethod( FindShortGeneratorsOfSubgroup, "without option rec or func, permutation group method", [ IsPermGroup, IsPermGroup ], function(G,U) local S; S := StabilizerChain(U); return FindShortGeneratorsOfSubgroup(G,U, rec( membershiptest := function(a,b) return SiftGroupElement(S,a).isone; end, sizetester := function(a) return Size(StabilizerChain(a)); end )); end ); InstallMethod( FindShortGeneratorsOfSubgroup, "without option rec or func, matrix group method", [ IsMatrixGroup, IsMatrixGroup ], function(G,U) local S; S := StabilizerChain(U); return FindShortGeneratorsOfSubgroup(G,U, rec( membershiptest := function(a,b) return SiftGroupElement(S,a).isone; end, sizetester := function(a) return Size(StabilizerChain(a)); end )); end );
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2026-10-13