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Quellcodebibliothek Statistik Leitseite products/Sources/formale Sprachen/GAP/grp/ (GAP Algebra Version 4.15.1^©) Datei vom 18.9.2025 mit Größe 198 kB

Quelle grpperm.gi Sprache: Shell

#############################################################################
##
## This file is part of GAP, a system for computational discrete algebra.
## This file's authors include Heiko Theißen.
##
## Copyright of GAP belongs to its developers, whose names are too numerous
## to list here. Please refer to the COPYRIGHT file for details.
##
## SPDX-License-Identifier: GPL-2.0-or-later
##

#############################################################################
##
#M CanEasilyTestMembership( <permgroup> )
##
InstallTrueMethod

#############################################################################
##
#M CanEasilyComputePcgs( <permgroup> )
##
## solvable permgroups can compute a pcgs. (The group may have been told it
## is solvable without actually doing a solvability test, so we need the
## implication.)
java.lang.StringIndexOutOfBoundsException: Index 2 out of bounds for length 0

#M CanComputeSizeAnySubgroup
java.lang.StringIndexOutOfBoundsException: Index 5 out of bounds for length 0

#############################################################################
##
#M AsSubgroup( <G>, ) . . . . . . . . . . . . with stab chain transfer
##
InstallMethodjava.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0
IsIdenticalObj IsPermGroup : []java.lang.StringIndexOutOfBoundsException: Index 20 out of bounds for length 20
 function( G, U )
 ##
 # test if the parent is already alright#M IndependentGeneratorsOfAbelianGroup( <G> ) . . . . . . . nice generatorsj];
 if IsPermGroupandIsAbelian ,

 fi;

 if not IsSubsetinds
return java.lang.StringIndexOutOfBoundsException: Index 18 out of bounds for length 18
 fi;
 S:= SubgroupNC ,# generators of -Sylowsubgroup
 UseIsomorphismRelation( U, S g od;
tRelation ,S ;
 if java.lang.StringIndexOutOfBoundsException: Index 12 out of bounds for length 0
 one=G;
 fi;
 S
 p>1

#############################################################################
##
#M CanEasilyComputeWithIndependentGensAbelianGroup( <permgroup> )
##
 for g GeneratorsOfGroup java.lang.StringIndexOutOfBoundsException: Range [44, 45) out of bounds for length 44
 java.lang.StringIndexOutOfBoundsException: Range [40, 39) out of bounds for length 43

#############################################################################
##
#F IndependentGeneratorsAbelianPPermGroup( , ) . . . nice generators
##
InstallGlobalFunctionif( then g;;
 function ( P ;
 localtrnsb =[java.lang.StringIndexOutOfBoundsException: Range [0, 36) out of bounds for length 21
 pows, # their powers
 base, # the base of the vectorspace
 # the size of the group generated by <inds>
 orbs, # orbits
 ;
 gens
 one
 ,java.lang.StringIndexOutOfBoundsException: Range [24, 23) out of bounds for length 26
 exp return inds; Add [] ^(hk))java.lang.StringIndexOutOfBoundsException: Index 50 out of bounds for length 50
##
#F OrbitPerms( <gens>, <d> ) . . . . . . . . .
 b, # basepoint
 c, # other point in orbit
 i, jjava.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0

 # initialize the list of independent generators
 inds := [ else
java.lang.StringIndexOutOfBoundsException: Index 15 out of bounds for length 15
base: ]
 size := 1;
 orbs return [ gens2, )java.lang.StringIndexOutOfBoundsException: Index 32 out of bounds for length 32
 trns := [];

# arethe forremaining
 gens= GeneratorsOfGroup(
 one

 # loop over the exponents# apply all generators <gen>
exp=(( g- ,p ;
 for i in [exp,exp-1..1] do

 # loop over the remaining generators
 : ]
 [mg] then
 g := gens[j];
 =g p^i);

 # reduce <g> and <h>
 nothjava.lang.StringIndexOutOfBoundsException: Index 30 out of bounds for length 30
#############################################################################
 do
 g := g / pows[ trns##
 h :h S(h ;
 od;

 p, factor of group group
 if the max by<>
 Add( inds, dom:([.max] )java.lang.StringIndexOutOfBoundsException: Index 40 out of bounds for length 40
 Add
 Add(, h );
 fst := Position #
 b := SmallestMovedPoint(h);
 # loop over all primes
 inds =[java.lang.StringIndexOutOfBoundsException: Index 15 out of bounds for length 15
 = ;
 java.lang.StringIndexOutOfBoundsException: Index 19 out of bounds for length 19
 ino in java.lang.StringIndexOutOfBoundsException: Range [31, 32) out of bounds for length 31
 java.lang.StringIndexOutOfBoundsException: Range [16, 1) out of bounds for length 36
 Add( :pnt^ gen;
 java.lang.StringIndexOutOfBoundsException: Index 25 out of bounds for length 0
 od;
 od;

 # otherwise reduce and add to gens2
 else
 java.lang.StringIndexOutOfBoundsException: Range [26, 19) out of bounds for length 37

 fi;g: g^ojava.lang.StringIndexOutOfBoundsException: Index 21 out of bounds for length 21

 od;

 # prepare for the next round
 gens := gens2;
 pows domimg

 od;

 # return the independent generators
 return inds;
end;

#############################################################################
##
#M IndependentGeneratorsOfAbelianGroup( <G> ) . . . . . . . nice generators
##
InstallMethod =Position , java.lang.StringIndexOutOfBoundsException: Index 46 out of bounds for length 46
 [ # add > are fixed
 function()
 local java.lang.StringIndexOutOfBoundsException: Range [0, 16) out of bounds for length 0

 p, return ;
 gens,java.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0
 # one generator
 ;# its order

 # loop over all primes
 inds := [];
 java.lang.StringIndexOutOfBoundsException: Index 6 out of bounds for length 2
 for p java.lang.StringIndexOutOfBoundsException: Range [0, 13) out of bounds for length 0
Orderg ))
 if p <> 1local max,orb n)then

 # compute the generators for the Sylow subgroupmaxif then
 gens: ]java.lang.StringIndexOutOfBoundsException: Index 19 out of bounds for length 19
forginG )do
 o
 while modp java.lang.StringIndexOutOfBoundsException: Index 7 out of bounds for length 7
 g: g;
if ;java.lang.StringIndexOutOfBoundsException: Index 54 out of bounds for length 54
 od;

 # append the independent generators for the Sylow subgroup
 Append( inds(SmallestMovedPoint
 IndependentGeneratorsAbelianPPermGroup(
 GroupByGenerators( gens, one ), p ) );

 fi
 od;

independentgenerators by increasing order
 ( , ;

 return the independent generators
 java.lang.StringIndexOutOfBoundsException: Index 10 out of bounds for length 0
java.lang.StringIndexOutOfBoundsException: Index 3 out of bounds for length 2

#############################################################################
##
#F OrbitPerms( <gens>, <d> ) . . . . . . . . . . . . . orbit of permutations
##
InstallGlobalFunction( OrbitPerms,
 java.lang.StringIndexOutOfBoundsException: Index 9 out of bounds for length 0

 # get the largest point <max> moved by the group <G>
 =LargestMovedPoint( );

 # handle fixpoints
 if 1.max] then
 return [ d ];
 fijava.lang.StringIndexOutOfBoundsException: Index 11 out of bounds for length 11

 # start with the singleton orbit
 orb := [ d ];
java.lang.StringIndexOutOfBoundsException: Index 63 out of bounds for length 47
 new[d] := false;

 # loop over all points found
 for pnt in orb do;

 # apply all generators <gen>
 in do
^java.lang.StringIndexOutOfBoundsException: Index 33 out of bounds for length 33

 # add the image <img> to the orbit if it is new
 if new[[and ] 0,
 Add( java.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0
 new[img] := false;
 ,java.lang.StringIndexOutOfBoundsException: Range [47, 47) out of bounds for length 15

 od;

 od;
 java.lang.StringIndexOutOfBoundsException: Range [15, 14) out of bounds for length 19
;

#############################################################################
##
#F OrbitsPerms( <gens>, <D> ) . . . . . . . . . . . orbits of permutations
##
InstallGlobalFunction( OrbitsPerms, function( gens, ^ gen <> java.lang.StringIndexOutOfBoundsException: Range [0, 35) out of bounds for length 6
 local max dom,new, ,fst, orb,


 max := LargestMovedPoint( gens );
 dom := BlistList( [1..java.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0
 new := BlistList( [ ;
 ;

 # repeat until the domain is exhausted
 fst := Position( Set )java.lang.StringIndexOutOfBoundsException: Index 22 out of bounds for length 22
 while fst >fail do

 # start with the singleton orbit new:B .][.];
 (java.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0
InstallMethod( MovedPoints
 # start with the singleton orbit


 for;

 #############################################################################
 forgeningensdo
 :


Add )
 new[img: pnt java.lang.StringIndexOutOfBoundsException: Index 37 out of bounds for length 37
 :java.lang.StringIndexOutOfBoundsException: Range [20, 19) out of bounds for length 24
 fi;

 ,

 od;

orbittothe listof java.lang.StringIndexOutOfBoundsException: Range [53, 49) out of bounds for length 69
 Addforanempty list new[]=false;
 fst and ]

 od;

 # add the remaining points of <D>, they are fixed
 for##
 if
 Add( orbsjava.lang.StringIndexOutOfBoundsException: Range [14, 13) out of bounds for length 33
 fi;
 od;

 return orbs;
java.lang.StringIndexOutOfBoundsException: Range [12, 1) out of bounds for length 46

#############################################################################
##
#M SmallestMovedPoint( <C> ) . . . . . . . for a collection of permutations
##
BindGlobaljava.lang.StringIndexOutOfBoundsException: Index 33 out of bounds for length 24
 ,

 min :,0
 for gen in C fi;
e
 #############################################################################
 if m MovedPoints(( <G>) .... . .##
 min := m;
 fi;
fi
 IsPermGroup ,0#java.lang.StringIndexOutOfBoundsException: Range [77, 2) out of bounds for length 77
 return min;
endG>MovedPoints (G ;

InstallMethod( SmallestMovedPoint,
 "for a java.lang.StringIndexOutOfBoundsException: Range [0, 21) out of bounds for length 0
 true,
 [ #M NrMovedPoints( <G> )
intPerms##

InstallMethod( SmallestMovedPoint,
 list,
 true,
 [truejava.lang.StringIndexOutOfBoundsException: Index 9 out of bounds for length 9
 - );

#############################################################################
##
#M LargestMovedPoint( <C> ) . . . . . . . for a collection of permutations
##
BindGlobal( "LargestMovedPointPerms", function(G)
 local max, m, gen;

 max =0java.lang.StringIndexOutOfBoundsException: Index 13 out of bounds for length 13
 forList(,IsRange;# save memory if long orbits
ifnot IsOne(gen ) then
 m := LargestMovedPointPerm( i java.lang.StringIndexOutOfBoundsException: Range [15, 16) out of bounds for length 15
 ,)
 max := m;
 java.lang.StringIndexOutOfBoundsException: Index 15 out of bounds for length 15
 fi;
 od;
 return max;
nd)

InstallMethod<H> ) . . . . . . . . . . . test if two perm groups are equal
 "for a java.lang.StringIndexOutOfBoundsException: Index 16 out of bounds for length 2
 truejava.lang.StringIndexOutOfBoundsException: Index 9 out of bounds for length 9
 [ IsPermCollection
 LargestMovedPointPerms );

InstallMethod(LargestMovedPoint
 "java.lang.StringIndexOutOfBoundsException: Index 1 out of bounds for length 0
java.lang.StringIndexOutOfBoundsException: Index 9 out of bounds for length 9
 [ IsList andfi
 LargestMovedPointPerms );

#############################################################################
##
#F MovedPoints( <C> ) . . . . . . . . . . for a collection of permutations
##
InstallGlobalFunction(MovedPointsPerms,java.lang.StringIndexOutOfBoundsException: Index 42 out of bounds for length 0
 local mov, gen, pnt;

 : [ ;
 for C do
 java.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0
 for max :=java.lang.StringIndexOutOfBoundsException: Index 13 out of bounds for length 13
 LargestMovedPoint gen ] do
 if ;
 mov notIsOne ) then

 od;
 fi;
 od;
 return Set( mov );
end);

InstallMethod( ##
 [IsPermCollection0,
 fora max m then

 ] 0java.lang.StringIndexOutOfBoundsException: Index 23 out of bounds for length 23
 [ IsList and IsEmpty ], 0,
 MovedPointsPerms );

InstallMethod( MovedPoints#############################################################################fi;
function(p)
 return MovedPointsPerms([p]);return#java.lang.StringIndexOutOfBoundsException: Index 2 out of bounds for length 2
"for java.lang.StringIndexOutOfBoundsException: Range [23, 22) out of bounds for length 30

#############################################################################
##
#M NrMovedPoints( <C> ) . . . . . . . . . for a collection of permutations
##
BindGlobal( "NrMovedPointsPerms", function( C )
 java.lang.StringIndexOutOfBoundsException: Range [0, 9) out of bounds for length 2

 mov := 0;
 sma: java.lang.StringIndexOutOfBoundsException: Index 11 out of bounds for length 2
 if sma )
 local elmelm ,java.lang.StringIndexOutOfBoundsException: Range [0, 26) out of bounds for length 9
;
 for pnt in [ sma )
 ############################################################################# - 1java.lang.StringIndexOutOfBoundsException: Index 19 out of bounds for length 19
>pnt
 : java.lang.StringIndexOutOfBoundsException: Index 28 out of bounds for length 28
 ;
 fipos: ( pos,len)
 ;
 odjava.lang.StringIndexOutOfBoundsException: Index 7 out of bounds for length 7
 mov;
end Error ">,n ]must ovedPoint( .

InstallMethod ;
 " returnelm
 [ IsPermCollection ],
 NrMovedPointsPermsjava.lang.StringIndexOutOfBoundsException: Range [25, 10) out of bounds for length 38

(NrMovedPoints
 "for an empty list fi;
 [ IsList and IsEmpty ],
 NrMovedPointsPerms );

#############################################################################
##
#M LargestMovedPoint( <G> ) . . . . . . . . . . . . for a permutation group
##
S:enum.tabChainjava.lang.StringIndexOutOfBoundsException: Index 25 out of bounds for length 25
 " a permutationgroup"
 true,
 [IsPermGroup ] ,
 G -MovedPointsPerms ;

#############################################################################
##
#M SmallestMovedPoint( <G> ) . . . . . . . . . . . . for a permutation group
##
(SmallestMovedPoint
 "or a permutation group",
 true,
 [ IsPermGroup ], 0,
G- (GeneratorsOfGroup

#############################################################################
##
#M MovedPoints( <G> ) . . . . . . . . . . . . . . . for a permutation group
##
InstallMethod( MovedPoints,
 " fail
 true,
 [ java.lang.StringIndexOutOfBoundsException: Index 9 out of bounds for length 7
 G -> MovedPointsjava.lang.StringIndexOutOfBoundsException: Index 7 out of bounds for length 7

#############################################################################
##
#M NrMovedPoints( <G> ) . . . . . . . . . . . . . . for a permutation group
##
InstallMethod(functionjava.lang.StringIndexOutOfBoundsException: Index 11 out of bounds for length 11
 "a permutation "
 true,
 [ IsPermGroup ], 0,
 - =java.lang.StringIndexOutOfBoundsException: Range [30, 29) out of bounds for length 35

#############################################################################
##
#M OrbitsMovedPoints( <G> )
##
(java.lang.StringIndexOutOfBoundsException: Range [34, 32) out of bounds for length 75
function(G)
,java.lang.StringIndexOutOfBoundsException: Index 10 out of bounds for length 10
 ojava.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0
oSjava.lang.StringIndexOutOfBoundsException: Index 17 out of bounds for length 17
o)# save memory if long orbits
 java.lang.StringIndexOutOfBoundsException: Index 28 out of bounds for length 0
 MakeImmutable(o)=java.lang.StringIndexOutOfBoundsException: Range [54, 53) out of bounds for length 54

 SetIsSSortedList(i,true);
 od;
 SetIsSSortedList(o,true);
 o
end);

#############################################################################
##
#M \=( <G>, <H> ) . . . . . . . . . . . test if two perm groups are equal
##
[S:[java.lang.StringIndexOutOfBoundsException: Index 19 out of bounds for length 19
function ( p:(S.,;
 # avoid new stabilizer chains if they are not computed yet
 java.lang.StringIndexOutOfBoundsException: Range [6, 4) out of bounds for length 9

 if not (HasGeneratorsOfGroup(#S=e,
 TryNextMethod ;
 ;

 or
 return##
 fi#M Iterator( <G> ) . . . . . . . . . ##

 if GeneratorsOfGroup(G) then
 return true;
 fi;

 # try to use one existing stab chainfor a group,
 H java.lang.StringIndexOutOfBoundsException: Range [53, 52) out of bounds for length 60
oupG,java.lang.StringIndexOutOfBoundsException: Range [64, 62) out of bounds for length 74
 fi;
 return #M MovedPoints( <G> ) . . . . . . . . . . . . . . . java.lang.StringIndexOutOfBoundsException: Index 55 out of bounds for length 2
 truejava.lang.StringIndexOutOfBoundsException: Index 9 out of bounds for length 9

#############################################################################
##
#M BaseOfGroup( <G> )
##
( BaseOfGroup,
 " "for apermutation "java.lang.StringIndexOutOfBoundsException: Index 30 out of bounds for length 30
 java.lang.StringIndexOutOfBoundsException: Index 9 out of bounds for length 9
 [ IsPermGroup fjava.lang.StringIndexOutOfBoundsException: Range [9, 8) out of bounds for length 11
ainMutable(G )))java.lang.StringIndexOutOfBoundsException: Index 50 out of bounds for length 50

#############################################################################
##
#M Size( <G> ) . . . . . . . . . . . . . . . . . . size of permutation group
##

 "#############################################################################
 true,#M Random( <G> ) . . . . . . . . . . . . . . . . . . . . . . random element

 G -> SizeStabChain( "for a random source andgroup,

#############################################################################
##
#M Enumerator( <G> ) . . . . . . . . . . . . enumerator of permutation group
##
BindGlobal( "java.lang.StringIndexOutOfBoundsException: Index 30 out of bounds for length 20
 function( enum, pos )
 function ( G function ( G, H ( java.lang.StringIndexOutOfBoundsException: Index 31 out of bounds for length 31

 S := enum!.stabChain;
 elm .java.lang.StringIndexOutOfBoundsException: Range [22, 21) out of bounds for length 22
 njava.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0
 pos := pos - 1;
 while java.lang.StringIndexOutOfBoundsException: Index 15 out of bounds for length 7
 len ;
 : java.lang.StringIndexOutOfBoundsException: Range [28, 27) out of bounds for length 50

 pos := QuoInt( pos, len );
 stabilizer
 od;

 ifif ajava.lang.StringIndexOutOfBoundsException: Range [23, 22) out of bounds for length 49
 < ]have an " ;
 fi;
;
end );

"
 function
 local ;

en
 java.lang.StringIndexOutOfBoundsException: Range [13, 12) out of bounds for length 18
 fijava.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0

 #M ClosureGrjava.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0
## The following function implements the method and may be called even with
 S := enum!.stabChain;
while java.lang.StringIndexOutOfBoundsException: Index 30 out of bounds for length 27
 : SorbitP java.lang.StringIndexOutOfBoundsException: Range [40, 39) out of bounds for length 47
 #pos := pos + val * ( Position( S.orbit, img ) - 1 );
 if##
 pos := pos + val * S.orbitpos[img];
 #val := val * Length( S.orbit );
 val := val * S.ol;
 = , >notin java.lang.StringIndexOutOfBoundsException: Index 50 out of bounds for length 50
 S := java.lang.StringIndexOutOfBoundsException: Range [4, 1) out of bounds for length 7

 # is the closure
 fi;
 od;G >( ForAll(gens,i->ForAll(GeneratorsOfGroup)->^i G)
 :g[#############################################################################
 ;
 fi;
 return pos;
end )

InstallMethod( Enumerator,
 :(Og)[)>sOneg^)
 [ IsPermGroup ] l:o;
function(G)
java.lang.StringIndexOutOfBoundsException: Index 1 out of bounds for length 0
 options.:=.java.lang.StringIndexOutOfBoundsException: Index 34 out of bounds for length 34
 S: :posjava.lang.StringIndexOutOfBoundsException: Index 12 out of bounds for length 12
 :[;
#
 e:Orbits : java.lang.StringIndexOutOfBoundsException: Range [31, 30) out of bounds for length 33
 :e;
 i:=Position(i = ;
 Add(p,e[i] =java.lang.StringIndexOutOfBoundsException: Range [26, 25) out of bounds for length 37
 =i[;
 od;
 if inpar

S(rec(:;
 ejava.lang.StringIndexOutOfBoundsException: Index 7 out of bounds for length 7
 ElementNumber :=: java.lang.StringIndexOutOfBoundsException: Range [50, 49) out of bounds for length 57
 :

 PrintObj := function( G # if DefaultStabChainOptions.tryPcgs
 ( "<enumerator of perm group># g -> g ^ gen in G ) ) then
 # Concatenation( GeneratorsOfGroup( G ), gens ) );Concatenation( GeneratorsOfGroup( G ), gens ) );

 # fi;
 while LengthS. if IsPcgs( pcgs ) then
 S.ol:=Length(S# if inpar then C := GroupStabChain( Pfi;
 S.orbitpos:val :=;
 for i in [1..Maximum(S.orbit)] do
 ion( C,G);
 if p<>fail then
 # fi;
 #
 od;
 S:=S.chain := Structu( StabChainMutable(G));
 od;
 return e;
);

#############################################################################
##
#M Iterator( <G> ) . . . . . . . . . . . . . . iterator of permutation group
##
InstallMethod( Iterator,
 "for a trivial permutation group",
java.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0

 return IteratorList([()]);
end);

InstallMethod( Iterator#val := val * Length( S.orbit ); java.lang.StringIndexOutOfBoundsException: Range [32, 30) out of bounds for length 42
ifjava.lang.StringIndexOutOfBoundsException: Range [34, 32) out of bounds for length 68
 [ S := S.stabilizer;
function( java.lang.StringIndexOutOfBoundsException: Index 11 out of bounds for length 11
 return IteratorStabChain(StabChain(G));

#############################################################################
##
#M Random( <G> ) . . . . . . . . . . . . . . . . . . . . . . random element
##
InstallMethodWithRandomSource( Random,
 random
 [ else=chain; ;
 function ,G)
 local C: Pnewgens;

 go down the stabchain and multiply random representatives
S=StabChainMutable G );
 rnd := S.identity;
bels ) <> 0 do
 rnd= LeftQuotient;
 Random(random: .random));
 S := S # get a good base
 odod

 # return the random element
return;
end );

#############################################################################
##
#M <g> in <G> . . . . . . . . . . . . . . . . . . . . . . . membership test
##
InstallMethod( \in,
 "for a permutation, and a InstallOtherMethod(,"java.lang.StringIndexOutOfBoundsException: Range [41, 40) out of bounds for length 52
 true,
 [ IsPerm, IsPermGroup
 function permgroup ,,
( G gin(G )
 return true;
 else
 G : StabChainMutable( G );
 return java.lang.StringIndexOutOfBoundsException: Index 24 out of bounds for length 16
 fi;
end

#############################################################################
##
#M ClosureGroup( <G>, <gens>, <options> ) . . . . . . closure with options
##
## The following function implements the method and may be called even with
## incomplete (temp) stabilizer chains.
",( , , java.lang.StringIndexOutOfBoundsException: Index 56 out of bounds for length 56
 local C, # closure of < <G>, <obj> >, result
java.lang.StringIndexOutOfBoundsException: Index 53 out of bounds for length 47
 g, # an element of gens

 newgens, # new generator list
 chain; # the stabilizer chain created

);# options will be overwritten
 # if all generators are in <G>, <G> is the closure ,IsRecord] 0,
 gens return );
 if IsEmpty( genslabels)<> do

 fi;

 # is the closure normalizing?
 if Length(gensInstallOtherMethod ,"mpty "truejava.lang.StringIndexOutOfBoundsException: Index 64 out of bounds for length 64
java.lang.StringIndexOutOfBoundsException: Range [13, 11) out of bounds for length 13
 g#M p:=PositionS.rbiti;
 if Size##
 o= if<fail
 options.limit:=Size(G)*o;
 else
 :FirstDifferenceOrderg[])-IsOnegj))java.lang.StringIndexOutOfBoundsException: Index 70 out of bounds for length 70
 options.limit:=o;
fi
 options # options record for stabilizer chain construction
 #Print(options.limit,"<<\n");
 gensN generators of the group <N>

 # otherwise decide between random and deterministic methods
 P := Parent( G );
 inpar := IsSubset( P, gens ,# values measuring randomness of <chain>
 notinparand not IsIdenticalObj( P, Parent( P ) ) do
 P := Parent( P );
 inpar := IsSubset Group( G );
 : One ;
 ifinparthen
 CopyOptionsDefaults
 elif not IsBound( options.randomN: SubgroupNC(unction(G)
 options. ([)java.lang.StringIndexOutOfBoundsException: Index 30 out of bounds for length 30
 fi;

G in <> with factorgroup

#AH 5-feb-96: Disabled (see gap-dev discussion).
# if DefaultStabChainOptions.tryPcgs
# and ForAll( gens, gen -> ForAll( GeneratorsOfGroup( G ),
# g -> g ^ gen in G ) ) then
# if inpar then
# C := SubgroupNC( P,
# Concatenation( GeneratorsOfGroup( G ), gens ) );
# else
# C := GroupByGenerators
# ( Concatenation( GeneratorsOfGroup( G ), gens ) );
# fi;
# pcgs:=TryPcgsPermGroup( [ C, G ], false, false, false );
# if IsPcgs( pcgs ) then
# chain:=pcgs!.stabChain;
# if inpar then C := GroupStabChain( P, chain, true );
# else C := GroupStabChain( chain ); fi;
# SetStabChainOptions( C, rec( random := options.random ) );
#
# UseSubsetRelation( C, G );
# return C;
# fi;
# fi;

 of G compatible with options.base;
 chain = if not IsEmpty)
 if IsBound( options. =ClosureGroup ,gensN options ;
 ChangeStabChain( chain, options.java.lang.StringIndexOutOfBoundsException: Range [0, 44) out of bounds for length 27
 java.lang.StringIndexOutOfBoundsException: Index 1 out of bounds for length 0
 fi;

 java.lang.StringIndexOutOfBoundsException: Range [11, 9) out of bounds for length 51
 gens ) ) <= 100 then
 options : :=S.identity;
 options.base := BaseStabChain( chain );
 for closure
if SiftedPermutation( chain, g ) <> chain.identity then
 StabChainStrong( chain, [if in
 fi
 od;
 else
 o:=ValueOption( fi;
 if IsInt(o) then options.limit odjava.lang.StringIndexOutOfBoundsException: Index 11 out of bounds for length 11
 chain : #
 fi;

 newgensif (chain.orbits )then
 IsBound(chain.aux ) then
 if inpar then C
 else C := GroupStabChain
 else
 #############################################################################
 else C := Group
 SetStabChainMutable,)
 fi;
 SetStabChainOptions( C, rec( random := optionscorrect: .orrect;

 rchain= chain)java.lang.StringIndexOutOfBoundsException: Index 46 out of bounds for length 46
 return C;
end );

InstallOtherMethod( ClosureGroup,"permgroup, : (chain,00,,10/chain.diam,0
 true,
tilresult< ;
 DoClosurePrmGp);

InstallOtherMethod true

ReturnFirst else

java.lang.StringIndexOutOfBoundsException: Range [18, 13) out of bounds for length 55
[,IsPerm], 0java.lang.StringIndexOutOfBoundsException: Index 29 out of bounds for length 29
function( G, g )
,[ , )java.lang.StringIndexOutOfBoundsException: Index 43 out of bounds for length 43
end );

InstallMethodjava.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0
 [##
function( G, gensjava.lang.StringIndexOutOfBoundsException: Index 11 out of bounds for length 11
 ##
end ) chain,java.lang.StringIndexOutOfBoundsException: Range [48, 47) out of bounds for length 53

InstallOtherMethod( ClosureGroup, "permgroupBindGlobal("",unction( gens options )
 [ IsPermGroup,IsPerm,IsRecord , 0,
function( G, g, options )
 return DoClosurePrmGp( G, [ g ], options );
endP inpar #java.lang.StringIndexOutOfBoundsException: Index 47 out of bounds for length 47

InstallOtherMethod( ClosureGroup, "permgroup, permgroup, options", newgens,
 [ IsPermGroup, IsPermGroup,
function( G, H, options )
 return DoClosurePrmGp( G, GeneratorsOfGroup( H
end );

java.lang.StringIndexOutOfBoundsException: Index 8 out of bounds for length 7
[and, IsRecord,0java.lang.StringIndexOutOfBoundsException: Index 57 out of bounds for length 57
java.lang.StringIndexOutOfBoundsException: Range [12, 11) out of bounds for length 13

#############################################################################
##
#M NormalClosureOp( <G>, ) . . . . . . . . . . . . . . . . in perm group
##
BindGlobal(
 local N, # normal closure of in <G>, resultjava.lang.StringIndexOutOfBoundsException: Range [22, 21) out of bounds for length 30
 chain, # stabilizer chain for the result

 options, # options record for stabilizer chain construction( ,"of group",
 gensG, # generators of the group <G>
 genG, # one generator of the group <G>
 gensN, # generators of the group <N>
 genN, # one generator of the group <N>
 a ofU
 random ;
 , missing, correct result i java.lang.StringIndexOutOfBoundsException: Range [10, 9) out of bounds for length 33

 # get a set of monoid generators of <G>
 java.lang.StringIndexOutOfBoundsException: Index 36 out of bounds for length 36
 one:= One( G );

 # make a copy of the group to be closed
 N := SubgroupNC( InstallMethod(java.lang.StringIndexOutOfBoundsException: Range [30, 29) out of bounds for length 76
 function, java.lang.StringIndexOutOfBoundsException: Index 16 out of bounds for length 16
 UseSubsetRelation CopyOptionsDefaults (OnTuples(GeneratorsOfGroup(G) g ,One(G )java.lang.StringIndexOutOfBoundsException: Index 78 out of bounds for length 78
 SetStabChainMutable( N, StabChainMutable( U ) );
nOptionselifIsBound java.lang.StringIndexOutOfBoundsException: Range [30, 29) out of bounds for length 44
 if# perhaps <G> is C solvablejava.lang.StringIndexOutOfBoundsException: Range [56, 55) out of bounds for length 61
 else random :AH 5-java.lang.StringIndexOutOfBoundsException: Index 9 out of bounds for length 0
 options.tempbgroup( <U# g -> g ^ gen

 # make list of conjugates to be added to N# C := SubgroupNC( P, ,
 repeat
 gensN := [ ];
oriin[ 1.10] java.lang.StringIndexOutOfBoundsException: Range [33, 34) out of bounds for length 33
 genG := # ( Concatenation( GeneratorsOfGroup( G ), gens ) );
 cnj := SCRRandomSubproduct( Concatenation# pcgs:=TryPcgsPermGroup( [ C, G ], false, false, false );
 v, # random subproduct of V.generators
 if not cnjcomm, # commutator of u,v
 Add( gensN cnj );
 fi;
 od;
 if not IsEmptyjava.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0
 N := ClosureGroup( N, gensN, options );
 ;
 until IsEmpty( gensN );

 # Guarantee that all conjugates are in the normal closure: Loop over all
 # generators of N
 ifLength( GeneratorsOfGroup( U ) ) *
 for genN in gensN do

 loopover repeat
 for genG in gensG java.lang.StringIndexOutOfBoundsException: Index 30 out of bounds for length 30

 # make sure that the conjugated element is in the closure

 java.lang.StringIndexOutOfBoundsException: Range [30, 14) out of bounds for length 33
 N := ClosureGroup (optionsreduced ) and options. IsBound( options.reduced ) and options.reduced

 fi;

 od;

 od;

 list, random = 0java.lang.StringIndexOutOfBoundsException: Index 59 out of bounds for length 59
 chain (N )
 if not IsBound( chain.orbits ) then
 if IsBound( chain.aux ) doneCUV (,g)< .dentitythen
 chain:=CUV=ClosureGroup U,V;
 fi;
 result := chain.identity;
 fi;
 missing := chain.missing;
 (C.:andomjava.lang.StringIndexOutOfBoundsException: Index 73 out of bounds for length 73
 rchain := IsInt()then optionsl:o;fijava.lang.StringIndexOutOfBoundsException: Index 45 out of bounds for length 45
 result := VerifySGS( rchain, missing, correct );
 not IsPerm(result))then
 repeat
 result= SCRStrongGenTest2chain00,1,1010/hain.diam,0,];


 chain: [java.lang.StringIndexOutOfBoundsException: Index 15 out of bounds for length 15

 =Comm, ));
 ifjava.lang.StringIndexOutOfBoundsException: Range [17, 1) out of bounds for length 65
:[,00,;
 else
,4QuoInt(+,2)4java.lang.StringIndexOutOfBoundsException: Range [51, 14) out of bounds for length 14
 fi;
 <200 then
 param ;
 param[]: 2java.lang.StringIndexOutOfBoundsException: Index 26 out of bounds for length 26
 fi
 java.lang.StringIndexOutOfBoundsException: Range [34, 30) out of bounds for length 39
 chainDoClosurePrmGp);
 chaincorrect .missing);
 dentitycorrect java.lang.StringIndexOutOfBoundsException: Range [64, 65) out of bounds for length 64
result= , java.lang.StringIndexOutOfBoundsException: Range [54, 53) out of bounds for length 56
 fi;
 chain := java.lang.StringIndexOutOfBoundsException: Index 29 out of bounds for length 13
 java.lang.StringIndexOutOfBoundsException: Range [0, 6) out of bounds for length 0
 SetStabChainMutable( N, ##
 if result <> hainidentity then
 N := InstallMethod( DerivedSubgroup,"permgrps",true, [ IsPermGroupfunction(,g )
 fi;

 g # random subproducts of generators
StabChainOptionsN.andomnstallMethod ClosureGroup permgroup,elements",true,

 , list of commutators
 UseSubsetRelation N,U)java.lang.StringIndexOutOfBoundsException: Index 30 out of bounds for length 30
 return N;
end );

InstallMethodInstallOtherMethod( ClosureGroup, "permgroup, element, options",true,
 IsIdenticalObj, [IsPermGroupD TrivialSubgroup(G);
function
 # test applicability# if there are >4 generators, use random subproducts
if G,
 TryNextMethod();
 fi;
 return DoNormalClosurePermGroup java.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0
end);

#############################################################################
##
#M ConjugateGroup( <G>, <g> ) . . . . . . . . . . . . of permutation group
##
 : ;
function( G, g )
local H, S;

 nTuples( GeneratorsOfGroup G ), g), One( G) );
 if HasStabChainMutable(G) then
 S : Length(list 0
ConjugateStabChain (G ,S,g,g )java.lang.StringIndexOutOfBoundsException: Index 59 out of bounds for length 59
 InstallOtherMethod( ClosureGroup, "empty list and options",true,
 elif HasSize(G) then
 (H,(G))java.lang.StringIndexOutOfBoundsException: Index 25 out of bounds for length 25
 fi;
 UseIsomorphismRelation(G, ##
 return H; java.lang.StringIndexOutOfBoundsException: Range [18, 19) out of bounds for length 18
end )java.lang.StringIndexOutOfBoundsException: Index 5 out of bounds for length 2

#############################################################################
##
#M CommutatorSubgroup( , <V> ) . . . . . . . . . . . . . for perm groups
##
InstallMethod( CommutatorSubgroup, "permgroups", rchain ;
 [ java.lang.StringIndexOutOfBoundsException: Range [67, 26) out of bounds for length 67
 function( U, V () java.lang.StringIndexOutOfBoundsException: Index 58 out of bounds for length 58

 java.lang.StringIndexOutOfBoundsException: Range [12, 1) out of bounds for length 56
 doneCUV, # boolean; true if CUV is computed
 u, # random subproduct of U.generators
 v param, missing, correct, ,one
 comm, # commutator of u,v
 list, # get a set of monoid generators of <G>java.lang.StringIndexOutOfBoundsException: Index 12 out of bounds for length 12
 i; loopvariable

 # [ , <V> ] = normal closure of < [ , <v> ] >.
 = ())java.lang.StringIndexOutOfBoundsException: Index 51 out of bounds for length 51
 doneCUV :(, (=java.lang.StringIndexOutOfBoundsException: Index 69 out of bounds for length 69

 # if there are lot of generators, use random subproductsoptions : ShallowCopy
 Length ( )*
 Length( GeneratorsOfGroup

end
java.lang.StringIndexOutOfBoundsException: Index 40 out of bounds for length 30
 u := SCRRandomSubproduct( GeneratorsOfGroup##
 v := SCRRandomSubproduct
 comm :
 if not comm in C then
 Add( list, comm ) ;
 fi;
 od
 if Length(list) > genG: SCRRandomSubproduct( gensG, One( G ) );
 C := ClosureGroup( C## William M. Kantor,:=SCRRandomSubproductutation Groups of Degree $nl 106$,
 temp := GeneratorsOfGroup )(" ,true, ] 0,
java.lang.StringIndexOutOfBoundsException: Index 47 out of bounds for length 33
 CUV := ClosureGroup( U, V );
 doneCUV := true;
 fijava.lang.StringIndexOutOfBoundsException: Index 17 out of bounds for length 17
 # force closure test
 StabChainOptions if notIsEmpty(gensN ) then
 C := DoNormalClosurePermGroup( java.lang.StringIndexOutOfBoundsException: Index 51 out of bounds for length 51
 fi;
 until IsEmpty( return # Guarantee thatall :Loopover
 fi;

 # first find a transitive representation for <G>
 =[]java.lang.StringIndexOutOfBoundsException: Index 15 out of bounds for length 15
 for (U do
 for vin ionHomomorphism(G ,surjective")
 comm=Comm u v ;
 if not comm in then

 fi;
 od;
java.lang.StringIndexOutOfBoundsException: Index 25 out of bounds for length 7
 ifjava.lang.StringIndexOutOfBoundsException: Index 10 out of bounds for length 7
 C := ClosureGroup
 temp := true ) );
 if not doneCUV then
 CUV := ClosureGroup( U, V java.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0
 doneCUV := true;
 fi;
 # force closure test falsefi;
 StabChainOptions(C).random java.lang.StringIndexOutOfBoundsException: Index 7 out of bounds for length 7
C :=( CUV,C );
 fi;

 return C;
end );

#############################################################################
##
#M DerivedSubgroup( <G> ) . . . . . . . . . . . . . . of permutation group
##
InstallMethod( java.lang.StringIndexOutOfBoundsException: Index 30 out of bounds for length 11
 function( G )
 local D, # derived subgroup of <G>, result
 g, h, # random subproducts of generators
 comm, # their commutator
 list # list of commutators
 i,j; # loop variables

 # find the subgroup generated by the commutators of the generators
 D :=

e
 if Length( param := [k,4,0true;
 repeat
 list param: G 25920 java.lang.StringIndexOutOfBoundsException: Index 44 out of bounds for length 44
 for i fi;
 : SCRRandomSubproduct(GeneratorsOfGroup( G ),One)
 h = java.lang.StringIndexOutOfBoundsException: Range [60, 59) out of bounds for length 77
 or 21andSize()237783237120 java.lang.StringIndexOutOfBoundsException: Index 71 out of bounds for length 71
 if not result := or (d = 8192 and Size(G8191^3-8191/ # PSL(2,8191)
 ( comm)java.lang.StringIndexOutOfBoundsException: Index 37 out of bounds for length 37
 fi;
 ;
 java.lang.StringIndexOutOfBoundsException: Range [22, 21) out of bounds for length 36
 D : result:SCRStrongGenTest2 java.lang.StringIndexOutOfBoundsException: Range [54, 53) out of bounds for length 56
 temp := true ) );
 # force closure test

 java.lang.StringIndexOutOfBoundsException: Range [33, 32) out of bounds for length 71
 else
 StabChainOptions(D).randomfi
 fi;
 ()java.lang.StringIndexOutOfBoundsException: Range [32, 30) out of bounds for length 39
 fi;
 until list =
 fi


 listjava.lang.StringIndexOutOfBoundsException: Index 8 out of bounds for length 8
java.lang.StringIndexOutOfBoundsException: Range [46, 45) out of bounds for length 58
 java.lang.StringIndexOutOfBoundsException: Range [15, 14) out of bounds for length 25
 comm := Comm( GeneratorsOfGroup( G )[i],
 java.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0
 if not#M ConjugateGroup( <G>, <g[1092,43589145600,InstallMethod( ConjugateGroup, ", <g>", true, [ IsPermGroup, IsPerm ], 0,
 Add( ] ;
 fi;
 od;
 od;
 not (G
 [] 55java.lang.StringIndexOutOfBoundsException: Index 50 out of bounds for length 49
 temp := 168,java.lang.StringIndexOutOfBoundsException: Range [26, 25) out of bounds for length 34
 # give D a proper randomness
 if H;
 StabChainOptions(Djava.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0

 java.lang.StringIndexOutOfBoundsException: Range [16, 12) out of bounds for length 49
 ;

 D =java.lang.StringIndexOutOfBoundsException: Range [34, 18) out of bounds for length 70
 fi;

 if HasIsNaturalSymmetricGroup(G) and

 #java.lang.StringIndexOutOfBoundsException: Range [31, 30) out of bounds for length 37
end otherwise< ,

#############################################################################
##
#M IsSimpleGroup( <G> ) . . . . . . . test if a permutation group is simple
##
## This is a most interesting function. It tests whether a permutation
## group is simple by testing whether the group is perfect and then only
## looking at the size of the group and the degree of a primitive operation.
## Basically it uses the O'Nan--Scott theorem, which gives a pretty clear
## description of perfect primitive groups. This algorithm is described in
## William M. Kantor,
## Finding Composition Factors of Permutation Groups of Degree $n\leq 10^6$,
## J. Symbolic Computation, 12:517--526, 1991.
##
java.lang.StringIndexOutOfBoundsException: Range [7, 2) out of bounds for length 50
 function ( G )
 local #java.lang.StringIndexOutOfBoundsException: Range [36, 35) out of bounds for length 49
 homlocalsom,,new,li,,,b;
 d, # degree of <G>
 ,m # $d = n^m$:MovedPoints()java.lang.StringIndexOutOfBoundsException: Index 22 out of bounds for length 22
 simple, # list of orders of simple groups
 ifin java.lang.StringIndexOutOfBoundsException: Range [36, 37) out of bounds for length 36
 sjava.lang.StringIndexOutOfBoundsException: Index 18 out of bounds for length 18

 # if <G> is the trivial group, it is not simple
java.lang.StringIndexOutOfBoundsException: Index 42 out of bounds for length 27
 return false;
 fi;

 # first find a transitive representation for <G>
 D ,V )
 elvth=Product(1 =java.lang.StringIndexOutOfBoundsException: Index 35 out of bounds for length 34
 hom := ActionHomomorphism( G,
 if Size( G ) <> Size C:
 return false;
 fi;
 G := Image( hom );
 fi;

 # next find a primitive representation for <G>
 D=;
 while Length( for ( )java.lang.StringIndexOutOfBoundsException: Index 40 out of bounds for length 40
 hom := ctionHomomorphism hom := ActionHomomorphism( G
 if Size( G ) <> Size( Image( hom ) ) od
 return false;
 fi;
 G = ,java.lang.StringIndexOutOfBoundsException: Range [31, 30) out of bounds for length 33
 D := Blocks( java.lang.StringIndexOutOfBoundsException: Range [0, 22) out of bounds for length 7
 od; = java.lang.StringIndexOutOfBoundsException: Range [26, 25) out of bounds for length 53

 # compute the degree $d$ and express it as $d = n^m$
 D : G)java.lang.StringIndexOutOfBoundsException: Index 26 out of bounds for length 26
 d := Length( D );
 n := SmallestRootInt( d );
java.lang.StringIndexOutOfBoundsException: Index 39 out of bounds for length 24
 if 10^6 < d then
 Error"cannot decide else
 fi;

 # if $G = C_p$, it is simple
 if IsPrimeInt( Size( G ) failthenC = DoNormalClosurePermGroup(java.lang.StringIndexOutOfBoundsException: Range [13, 12) out of bounds for length 18
 true;

 # if $G = A_d$, it is simple (unless $d < 5$)
 elif end;
 return

 # if $G = S_d$, it is not simple (except $S_2$)
 elif IsNaturalSymmetricGroup(G) then
 return 2 = d;

# if $G$ is not perfect, it is not simple (unless $G = C_p$, see above)
 D IsPcgs(cgs java.lang.StringIndexOutOfBoundsException: Range [22, 23) out of bounds for length 22
 return false;

#java.lang.StringIndexOutOfBoundsException: Index 59 out of bounds for length 59
 elif Size( G ) = d ^ 2 then
 return false;

 # if $d$ is a prime, <G> is simple
 elif java.lang.StringIndexOutOfBoundsException: Index 12 out of bounds for length 0
 return true;

 # if $G = U(4,2)$, it is simple (operation on 27 points)
 elif d = 27 and Size( G ) = 25920 then
 return D := G)

 if $G = PSL(n,q)$, it is simple (operations on prime power points)
 elif ( (d = 8;
 or (d = 9 and Size(G) = (8^3-8) ) # PSL(2,8)
 or (d =##
 or (d = 121 and Size(G) = 237783237120) # PSL(5,3)
 or (d = 128 and Size(G) InstallMethodjava.lang.StringIndexOutOfBoundsException: Range [61, 59) out of bounds for length 77
 : g,hjava.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0
 or (d##
 or (d = 524288 and Sizem##
 and IsTransitive(
 then
 return true;

 # if $d$ is a prime power, <G> is not simple (except the cases above)
 elif d then
 return false;

 # if we don't have at least an $A_5$ acting on the top, <G> is simple
 java.lang.StringIndexOutOfBoundsException: Range [13, 8) out of bounds for length 21
 return true;

 # otherwise we must check for some special cases
 else

 # orders of simple subgroups of $S_n$ with primitive normalizer
 simple := [ ,#############################################################################
 [60,360],,,, # 5: A(5), A(6)#M PcgsPCentralSeriesPGroup( <G> )
 [ fi
 )
 java.lang.StringIndexOutOfBoundsException: Range [6, 5) out of bounds for length 11
 [360,2520,20160,653837184000 i .Length(( )) java.lang.StringIndexOutOfBoundsException: Range [0, 58) out of bounds for length 22
 ];

 # orders of transitive perfect subgroups of $S_m$

 java.lang.StringIndexOutOfBoundsException: Range [12, 11) out of bounds for length 20
 [60,360], # 6: A(5), A(6)
 [168,2520], # 7: PSL(3,2), A(7)
#T disabled.
 ];

 # test the special cases (Kantor's Lemma 3)
 for s in simple[n] if not IsEmpty( list ) the
 for t # InstallMethod( DerivedSeriesOfGroup,"for permgrp", true, [ IsPermGroup ], 0,
 if Size( G ) mod (t * s^m)
 and (( StabChainOptions#java.lang.StringIndexOutOfBoundsException: Index 47 out of bounds for length 47
 sIsSolvableGroupG) and IsSolvableGroup() # TryNextMethod();
 then
 (D.inot(pcgs)then
 fi;
 od;
 od;

 # otherwise <G> is simple
 return true;

 fi

end );

#############################################################################
##
#M IsSolvableGroup( <G> ) . . . . . . . . . . . . . . . . solvability test
##

# fast test for finding iteratively nontrivial commutators in a permutation group.
# if this goes on too long the group may not be solvable. This can be much faster
# for large degree groups than to use the full pcgs machinery (which uses the same
# idea but builds a pcgs on the way).
# The function returns `true` if the group has been proved to be *non*solvable.
BindGlobal## William M. Kantor,
local
 # a few moved points
 som:=MovedPoints(G);
 if Length(som) = 0 then SetIsTrivial(G,true); return fail; fi;
 ;
 if Length(som)>100 then
 som:=som{List([1..100],java.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0
 fi;

 elvth:=One(G);
 fct:=function(G)
 elvth:=(,java.lang.StringIndexOutOfBoundsException: Range [12, 1) out of bounds for length 69
 java.lang.StringIndexOutOfBoundsException: Range [11, 10) out of bounds for length 17
 end;

 java.lang.StringIndexOutOfBoundsException: Index 5 out of bounds for length 0
 # find one nontrivial commutator
 new:=ail;
 i:=1;
 while i<)java.lang.StringIndexOutOfBoundsException: Index 6 out of bounds for length 6

 while j<= java.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0
 :Orbit( G ;
 new:=Commg, andjava.lang.StringIndexOutOfBoundsException: Range [63, 62) out of bounds for length 63
 fi;
=j+
 od;
 i:=i+1;
 od;
 if new=fail :=PcgsElementaryAbelianSerieG)

 # now take commutators with conjugates to get down
 i:=1;
 repeat
 returnpcgsjava.lang.StringIndexOutOfBoundsException: Index 36 out of bounds for length 36
 j:=1;
 new:=fail;
 l:=[# ( fam if Size(G) <> Size(Image(hom# [ IsList ], 0,
 while # local pcgs;fi;
 =(l))fct(G);
 b:=Firstlb->ForAny( or ( HasIsSolvableGroup1 ] )
 if b<>fail then
 :=java.lang.StringIndexOutOfBoundsException: Range [0, 17) out of bounds for length 9
 else
 Add(l,a);
 fi;
 j:=j+1;
 od;
 if =PCentjava.lang.StringIndexOutOfBoundsException: Range [32, 31) out of bounds for length 82
 java.lang.StringIndexOutOfBoundsException: Index 18 out of bounds for length 18
 fi
 ifactors(G))
 until i>bound;
 return true;
end);

 # if $G = C_p$, it simple
function()
local pcgs;
 if return(;

 pcgs:=java.lang.StringIndexOutOfBoundsException: Index 13 out of bounds for length 0
 if IsPcgs##
 SetIndicesEANormalSteps(pcgs,pcgs!.java.lang.StringIndexOutOfBoundsException: Index 58 out of bounds for length 0
 SetIsPcgsElementaryAbelianSeries(pcgs,true);
 HasPcgs(G) then
 SetPcgs elif IsNaturalSymmetricGroup(G) then
 fi;
 if not HasPcgsElementaryAbelianSeries
 (Gp)java.lang.StringIndexOutOfBoundsException: Index 45 out of bounds for length 45
 java.lang.StringIndexOutOfBoundsException: Index 7 out of bounds for length 7
 return true;
 else

 :StabChainMutableS )java.lang.StringIndexOutOfBoundsException: Index 50 out of bounds for length 50
end);

#############################################################################
##
#M IsNilpotentGroup( <G> ) . . . . . . . . . . . . . . . . . nilpotency test
##
java.lang.StringIndexOutOfBoundsException: Range [46, 13) out of bounds for length 72
 G

#############################################################################
##
#M PcgsCentralSeries( <G> )
##
InstallOtherMethod( java.lang.StringIndexOutOfBoundsException: Range [46, 45) out of bounds for length 63
functionjava.lang.StringIndexOutOfBoundsException: Index 11 out of bounds for length 11
local or =andG=8191-/
 pcgs:=TryPcgsPermGroup ,# domain of operation of <G>
 =5242873524287 ,#
 SetIndicesCentralNormalSteps(pcgs,pcgs!f # action homomorphism of <G> on <O> or 
 ;
 if java.lang.StringIndexOutOfBoundsException: Range [0, 10) out of bounds for length 8
 java.lang.StringIndexOutOfBoundsException: Index 1 out of bounds for length 0
 fi;
 java.lang.StringIndexOutOfBoundsException: Index 5 out of bounds for length 5
 return pcgs
end);

#############################################################################
##
#M PcgsPCentralSeriesPGroup( <G> )
##

function(G)
local java.lang.StringIndexOutOfBoundsException: Range [11, 10) out of bounds for length 11
 pcgs:=TryPcgsPermGroup( G, true, true, Factors(
 if IsPcgs(pcgs) then
 SetIndicesPCentralNormalStepsPGroup(pcgs,pcgsS =S.
 java.lang.StringIndexOutOfBoundsException: Index 24 out of bounds for length 7
 fi;
 if IsPcgs(pcgs# handle full group
 SetPcgs(G,pcgs);
 ;
 return pcgs;
end;

#T AH, 3/26/07: This method triggers a bug with indices. It is not clear
#T form the description why TryPcgs... should also give a drived series. Thus
#T disabled.
# #############################################################################
# ##
# #M DerivedSeriesOfGroup( <G> ) . . . . . . . . . . . . . . . derived series
# ##
# InstallMethod( DerivedSeriesOfGroup,"for permgrp", true, [ IsPermGroup ], 0,
# function( G )
# local pcgs, series;
#
# if (not DefaultStabChainOptions.tryPcgs
# or ( HasIsSolvableGroup( G ) and not IsSolvableGroup( G ) ))
# and not (HasIsSolvableGroup(G) and IsSolvableGroup(G)) then
# TryNextMethod();
# fi;
#
# pcgs := TryPcgsPermGroup( G, false, true, false );
# if not IsPcgs( pcgs ) then
# TryNextMethod();
# fi;
# SetIndicesEANormalSteps(pcgs,pcgs!.permpcgsNormalSteps);
# series := EANormalSeriesByPcgs( pcgs );
# if not HasDerivedSubgroup( G ) then
# if Length( series ) > 1 then SetDerivedSubgroup( G, series[ 2 ] );
# else SetDerivedSubgroup( G, G ); fi;
# fi;
# return series;
# end );

#############################################################################
##
#M LowerCentralSeriesOfGroup( <G> ) . . . . . . . . . lower central series
##
InstallMethod( LowerCentralSeriesOfGroup,"
 function( G )
 local pcgs, java.lang.StringIndexOutOfBoundsException: Index 23 out of bounds for length 0

 java.lang.StringIndexOutOfBoundsException: Index 6 out of bounds for length 0
 return [G];
 fi;

 if not DefaultStabChainOptions.# fasttest forjava.lang.StringIndexOutOfBoundsException: Range [36, 35) out of bounds for length 82
 or G ) not ia the# java.lang.StringIndexOutOfBoundsException: Range [15, 14) out of bounds for length 79
 and not (HasIsNilpotentGroup(G) and IsNilpotentGroup(G)) then
 TryNextMethod();
 fi;

 pcgs =TryPcgsPermGroup( f := ActionHomomorphism :( ,,java.lang.StringIndexOutOfBoundsException: Range [47, 45) out of bounds for length 61
 if not IsPcgs( pcgs ) then
 TryNextMethod();

 SetIndicesCentralNormalSteps(pcgs,java.lang.StringIndexOutOfBoundsException: Index 41 out of bounds for length 38
 series := CentralNormalSeriesByPcgs( pcgs );
if (G
 if Length( series )java.lang.StringIndexOutOfBoundsException: Index 11 out of bounds for length 11
 else SetDerivedSubgroup( G, G ); fi;
 fi;
 return series =new
end );

InstallOtherMethod( while j<=10 and new=fail
[ IsPermGroup:l^)
 # we want this to come *before* the method for Pcgs computable groups
 ##
 -java.lang.StringIndexOutOfBoundsException: Range [14, 13) out of bounds for length 40
function(G)
local pcgs;
 pcgs:=PcgsElementaryAbelianSeries(G);
 if (a)java.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0
 TryNextMethod();
 fi;
 java.lang.StringIndexOutOfBoundsException: Range [24, 21) out of bounds for length 24
end);

#InstallOtherMethod( ElementaryAbelianSeries, fam -> IsIdenticalObj
# ( fam, CollectionsFamily( CollectionsFamily( PermutationsFamily ) ) ),
# [ IsList ], 0,
# function( list )
# local pcgs;
#
# if IsEmpty( list )
# or not DefaultStabChainOptions.tryPcgs
# or ( HasIsSolvableGroup( list[ 1 ] )
# and not IsSolvableGroup( list[ 1 ] ) ) then
# TryNextMethod();
## fi;
#
## pcgs := TryPcgsPermGroup( list, false, false, true );
# if not IsPcgs( pcgs ) then return fail;
# else return ElementaryAbelianSeries( pcgs ); fi;
#end );

#############################################################################
##
#M PCentralSeries( <G>, ) . . . . . . . . . . . . . . p-central series
##
InstallMethod(java.lang.StringIndexOutOfBoundsException: Index 54 out of bounds for length 54
function( G, p )
 local pcgsend)java.lang.StringIndexOutOfBoundsException: Index 5 out of bounds for length 5

 if##
 TryNextMethod();
 java.lang.StringIndexOutOfBoundsException: Range [0, 4) out of bounds for length 2
 pcgs:=PcgsPCentralSeriesPGroup(G);
 if not IsPcgs(java.lang.StringIndexOutOfBoundsException: Range [0, 20) out of bounds for length 7
java.lang.StringIndexOutOfBoundsException: Index 21 out of bounds for length 20
 fi;
 returndeg 78125
end) p: (colljava.lang.StringIndexOutOfBoundsException: Index 2 out of bounds for length 2

#############################################################################
##
#M SylowSubgroup( <G>, ) . . . . . . . . . . . . . . Sylow $p$-subgroup
##
InstallMethod( SetIndicesCentralNormalSteps.java.lang.StringIndexOutOfBoundsException: Range [65, 63) out of bounds for length 65
 java.lang.StringIndexOutOfBoundsException: Range [16, 15) out of bounds for length 31
function( G, p )
local S;
 java.lang.StringIndexOutOfBoundsException: Index 21 out of bounds for length 5
 # enforce solvable methods if we detected anew that the group is
# .
 return SylowSubgroupOpjava.lang.StringIndexOutOfBoundsException: Index 26 out of bounds for length 0
 fi;
 S
 S :=#M PcgsPCentralSeriesPGroup( <G> )
 SetIsNilpotentGroup( S, InstallOtherMethod( PcgsPCentralSeriesPGroup ifdeg>78125
 if Size(S) > 1 then
 SetIsPGroup( S, true );
 SetPrimePGroup( S, p );
 SetHallSubgroup(G, [p] L : ClosureSubgroup( L,Centralizer(G L ) )
 fi;
 return S;
end );

InstallGlobalFunction( SylowSubgroupPermGroup, function( G, p java.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0
 local S, fi;;##
 , power
 O, # one orbit of <G> in this domain
 , # blocks of the operation of <G> on <D>
 f,#T AH, 3/26/07: This method triggers a bug with indices. It is not clear
 T, p-
 g, g2, # one element of <G>
 C, C2;(

 # get the size of the -Sylow subgroup
: 1; q := 1; while)

 l : Options.
ifq= 1then
 od
 fi;
 # go down in stabilizers as long as possible
 S := StabChainMutable# if Length( series ) > 1 then SetDerivedSubgroup( G, series[ 2 ] );
 while Length( S.orbit )
 S## the orbits of Stab_G(pnt) on all points of the orbit pnt^G. (As not
 od;
 G := GroupStabChain( ## subgroup.)

 # handle full group
 if q = Size(fjava.lang.StringIndexOutOfBoundsException: Range [13, 12) out of bounds for length 21
 return G;
 fi;

 # handle -Sylow subgroups of size 
 if q = p then
 MovedPoints)
 g: g ^Order( # orbit:=Difference(StabChainMu(.[java.lang.StringIndexOutOfBoundsException: Range [56, 55) out of bounds for length 58
 return SubgroupNC( G, [ g ] );
 fi;

java.lang.StringIndexOutOfBoundsException: Range [26, 25) out of bounds for length 33
 D := MovedPoints( G );
 if not IsTransitive( G, D ) then
 pcgs!;
 S:=G;
 D := ShallowCopy( java.lang.StringIndexOutOfBoundsException: Index 16 out of bounds for length 16
 while q < Size( S ) do
 O := Orbit( S, D[1] );
 f := ActionHomomorphism( S, O,"surjective" );
 T: Range(f, java.lang.StringIndexOutOfBoundsException: Range [16, 15) out of bounds for length 29
 :, java.lang.StringIndexOutOfBoundsException: Index 38 out of bounds for length 38
 SubtractSet( D, O );
 od;
 return S;
 fi;

 # if the group is not primitive work in the image firstjava.lang.StringIndexOutOfBoundsException: Range [19, 18) out of bounds for length 30
 B = Blocks ,D )java.lang.StringIndexOutOfBoundsException: Index 24 out of bounds for length 24
 if Length( B ) <> fijava.lang.StringIndexOutOfBoundsException: Index 5 out of bounds for length 5
 Info(InfoGroup,1,"PermSylow: blocks");
 f := ActionHomomorphism( G, B, OnSets,"surjective" );
 :=java.lang.StringIndexOutOfBoundsException: Range [43, 42) out of bounds for length 53
 if java.lang.StringIndexOutOfBoundsException: Index 15 out of bounds for length 11
 T := PreImagesSet(b=java.lang.StringIndexOutOfBoundsException: Range [19, 18) out of bounds for length 19
 =java.lang.StringIndexOutOfBoundsException: Range [40, 39) out of bounds for length 49
 return S
 fi;
 fi;

 # find a element whose centralizer contains a full -Sylow subgroup
 Info(InfoGroup,1,"PermSylow: element":oa;
 repeat g := Random( G ); until java.lang.StringIndexOutOfBoundsException: Index 40 out of bounds for length 32
 g = ^Orderg p);
 C := java.lang.StringIndexOutOfBoundsException: Index 15 out of bounds for length 11
 while GcdInt( q, Size( C ) ) < q =;
 repeat g2 := Random( C ); until Order( g2 ) mod:genjava.lang.StringIndexOutOfBoundsException: Range [25, 24) out of bounds for length 32
 g2 := g2 ^ (Order( if orbit[1]^gen<>orbi1^>java.lang.StringIndexOutOfBoundsException: Range [31, 30) out of bounds for length 57
 C2 := Centralizer( G, g2
 if#
 : =g2
 fi;
 wh (gen=java.lang.StringIndexOutOfBoundsException: Range [40, 39) out of bounds for length 49

 # the centralizer acts on the cycles of the element
 B := List( Cycles( g, D ), Set );
 Info(InfoGroup,1,"PermSylow: cycleaction");
 f := ActionHomomorphism( C, B, OnSets,"surjective" );
 T := SylowSubgroupPermGroup( =java.lang.StringIndexOutOfBoundsException: Range [21, 20) out of bounds for length 23
S : java.lang.StringIndexOutOfBoundsException: Range [22, 21) out of bounds for length 30
 return S;

end ;

#############################################################################
##
#M Socle( <G> ) . . . . . . . . . . . socle of primitive permutation group
##
InstallMethod#M AllBlocks . . . Representatives of all block systems
 function( G )
 local Omega, deg, coll ds L, z, ord,
 p, i;

 java.lang.StringIndexOutOfBoundsException: Range [24, 9) out of bounds for length 30

 if not IsPrimitive( G, Omega )# function( list )
 TryNextMethod();
 fi;

 # Affine groups first.
 L : , ;
 if L <> [] then
java.lang.StringIndexOutOfBoundsException: Index 8 out of bounds for length 1
 fi;

 deg := Length (pcgs thenreturnjava.lang.StringIndexOutOfBoundsException: Index 46 out of bounds for length 46
 if java.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0
 TryNextMethod();
 # the normal closure actually seems to be faster than derived series, so
 # this is not really a shortcut
java.lang.StringIndexOutOfBoundsException: Index 30 out of bounds for length 30
 # shortcut := true;
 if >java.lang.StringIndexOutOfBoundsException: Index 23 out of bounds for length 17
 # coll := Collected( Factors( deg ) );
 # d := Gcd( List( coll, c -> c[ 2 ] ) );
 # if d mod 5 = 0 then
 # m := 1;
 # for c in coll do
 # m := m * c[ 1 ] ^ ( c[ 2 ] / d );
 # od;
 # if m >= 5 then
 # shortcut := false;s[1)java.lang.StringIndexOutOfBoundsException: Index 27 out of bounds for length 27
 # fi;
 # fi;
 # fi;
 # if shortcut then
 # ds := DerivedSeriesOfGroup( G );
 # return Last(ds);
 # fi;
 fi;

 coll := ( "CreateAllCycleStructures, function(java.lang.StringIndexOutOfBoundsException: Index 51 out of bounds for length 51
 if deg < 78125l#
 p := Last(coll)[ 1 ];
 else
 i := Length( coll in do
 while coll[i [2 ] 1
 GG,( java.lang.StringIndexOutOfBoundsException: Index 50 out of bounds for length 50
 od;
 p := coll[ i m-]+
 fi;
 repeat
 -]
 z := java.lang.StringIndexOutOfBoundsException: Index 22 out of bounds for length 11
 ord := Order( z );
 Add(,m;
 od;
 until Index( G, Centralizerjava.lang.StringIndexOutOfBoundsException: Index 12 out of bounds for length 11

 O, one orbit of <G> in this domain
 # speed up the closure process
 L := NormalClosure,# action of<##
 deg =78125 then
l,,,g
 L := Last(ds);
 fi;
 if IsSemiRegular( L, Omega ) then
 L := ClosureSubgroup( L, Centralizer( G, L ) );
 java.lang.StringIndexOutOfBoundsException: Range [4, 1) out of bounds for length 29
 return L;
end );

#############################################################################
##
#M FrattiniSubgroup( <G> ) . . . . . . . . . . . . for permutation p-groups
##
 # go
 function( G )
 local p, l, k, iwhile Length .java.lang.StringIndexOutOfBoundsException: Range [26, 25) out of bounds for length 42

 if IsPGroup(G)then
 TryNextMethod(
 elif IsTrivial( G ) then
 return G fi;
 fi;
 p := PrimePGroup( q = p
 l = java.lang.StringIndexOutOfBoundsException: Index 32 out of bounds for length 32
 k := [ l[1]^p ];
 for i in [ 2 .. Length(l) ] do
 for j in [ 1 .. i-1 ] do
 if not IsTransitive( G, D ) then
 od;
 Addk [ii;
 od;
 return SolvableNormalClosurePermGroup =(D);
end );

#############################################################################
##
#M ApproximateSuborbitsStabilizerPermGroup(<G>,<pnt>) . . . approximation of
## the orbits of Stab_G(pnt) on all points of the orbit pnt^G. (As not
## all schreier generators are used, the results may be the orbits of a
## subgroup.)
##
InstallGlobalFunction( java.lang.StringIndexOutOfBoundsException: Index 33 out of bounds for length 17
 java.lang.StringIndexOutOfBoundsException: Range [59, 8) out of bounds for length 59
 local one, if java.lang.StringIndexOutOfBoundsException: Range [14, 13) out of bounds for length 29
 processStabGen, currPt,currGen, gens,f : G ,"java.lang.StringIndexOutOfBoundsException: Range [58, 57) out of bounds for length 61

 one:= One( G );
 if HasStabChainMutable( G ) = java.lang.StringIndexOutOfBoundsException: Range [41, 39) out of bounds for length 49
# if we already have a stab chain and bother computing the proper
# stabilizer, a trivial orbit algorithm seems best.
 return Concatenation([[punkt]],
 OrbitsDomain(G,),
 ()[));
# orbit:=Difference(StabChainMutable(G).orbit,[punkt]);
# stab:=Stabilizer(G,punkt));
# # catch trivial case
# if Length(stab)=0 then
# stab:=[ one ];
# fi;
# stgp:=1;
#
# processStabGen:=function()
# stgp:=stgp+1;
# if stgp>Length(stab) then
# StabGenAvailable:=false;
# fi;
# return stab[stgp-1];
# end;

 else
 # compute orbit and transversal
 Ggens:=GeneratorsOfGroup(G);
 orbit := [ punkt ]java.lang.StringIndexOutOfBoundsException: Index 22 out of bounds for length 7
 # the centralizer acts on the cycles of the element
 trans[ punkt ] := one;
 for in orbit do
 for gen in Ggens do
 img:=^;
 if not IsBound( trans[ img ] ) then
 Add( orbit, img );
 trans[img ] := gen;
 fi;
 od;
 odjava.lang.StringIndexOutOfBoundsException: Index 7 out of bounds for length 7

 fi;

 currPt:=1;
 currGen:=0;

 processStabGen:=functionGcdListcoll c>2]))
local ,enjava.lang.StringIndexOutOfBoundsException: Index 28 out of bounds for length 28
 if # := ;
 # next one
 currGen:=currGen+1;
 if currGen>Length
 currGen:1java.lang.StringIndexOutOfBoundsException: Index 21 out of bounds for length 21
 currPt);
 ifjava.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0
 return ()###############################################################################
 fi;
 fi;
 :currPtjava.lang.StringIndexOutOfBoundsException: Index 18 out of bounds for length 18
 b:()
 else ;
 a:=Random(1, Length(orbit));
 b:=Random(1, Length(Ggens));
 fi;
 # map a with b permutations
 img:=orbit[a]^java.lang.StringIndexOutOfBoundsException: Index 22 out of bounds for length 14
 if img=orbit[a] then
 return ();

 punkt:=orbit[a];
 :=Ggens[;
 while punkt<>orbit[1] do
 gen:=trans[punkt]*gen;
 java.lang.StringIndexOutOfBoundsException: Range [16, 15) out of bounds for length 36
 od;
 if ord = z ;
 punkt:= until mod =;
 while punkt<>orbit[1]z: ord od;
 gen:=gen/trans[punkt];
 punkt:=punkt/trans[punkt];
 od;
 if orbit[1]^>orbit1]ErrorUHEH";fi;
 fi;
 return gen;
 end;

 # as the first approximation take a subgroup generated by four nontrivial
 # Schreier generators
 changedL : Lastds);
 gens:=[];
ilechanged<=3 (orbit)dojava.lang.StringIndexOutOfBoundsException: Range [6, 5) out of bounds for length 5
 gen:=();
 while IsOne(gen) and currPt<=Length(orbit) do
 return java.lang.StringIndexOutOfBoundsException: Index 13 out of bounds for length 13
 od;
 Add(gens,gen);
 function
 ;
 if Length(gens)=0
orbit=
 else
 orbit:=OrbitsPerms(gens,orbit);
 fi;

 # todo: further fuse
 java.lang.StringIndexOutOfBoundsException: Range [6, 3) out of bounds for length 15
end );

#############################################################################
##
#M AllBlocks . . . Representatives of all block systems
##
icjava.lang.StringIndexOutOfBoundsException: Range [7, 6) out of bounds for length 47
function(g)
localgens,,ool,

 j
 local bl,bld,i,t,n;
 # find representatives of all potential suborbits
 bld:=List(Filtered(subo,i->not ForAny(b,j->if java.lang.StringIndexOutOfBoundsException: Range [44, 41) out of bounds for length 50
 bl:#M ApproximateSuborbitsStabilizerPermGroup(<G>,<pnt>) . . . approximation of
 if not IsPrimefi
 for i in bld do
 t:=Union(b,[i]);
java.lang.StringIndexOutOfBoundsException: Index 39 out of bounds for length 27
 if Lengthl:orAlljava.lang.StringIndexOutOfBoundsException: Range [60, 59) out of bounds for length 79
 not n[1]java.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0
 t:=n[1];
 (
 bl:=Concatenation(bl,[od
 fi;
 od;
 ;
 return bl;
 end;

 one:= One(#M GeneratorsSmallest Difference(G,[])))
 := (java.lang.StringIndexOutOfBoundsException: Range [40, 39) out of bounds for length 58
 if Length(gens( ( S# # catch trivial case
 subo:= # stab:=[ o ;
 Minimum(List(gens,java.lang.StringIndexOutOfBoundsException: Index 38 out of bounds for length 8
 dom:=Union#
 if not IsTransitive(g,bGen:=function()
 Error("AllBlocks, must be transitive");
 fi;
 pool:=[ ,engths)then
 return DoBlocks(subo[1]);
end

#############################################################################
##
#F SignPermGroup
##
InstallGlobalFunctionelse
 if ForAll(GeneratorsOfGroup(g),i->SignPerm(i)=1) then

 else
 java.lang.StringIndexOutOfBoundsException: Range [11, 10) out of bounds for length 14
 fi;
end );# get the sorted orbit (the basepoint will be the first point) : one;

BindGlobal( "CreateAllCycleStructures", function(n)
local i,j,l,m;
 l:[];
 for i Add(java.lang.StringIndexOutOfBoundsException: Index 20 out of bounds for length 0
 m:=]java.lang.StringIndexOutOfBoundsException: Index 10 out of bounds for length 10
 for jod;

 if IsBound(m[j-1]) then
 mjava.lang.StringIndexOutOfBoundsException: Index 12 out of bounds for length 12
 od;
 m[j- processStabGen:=#
 local plocal i,ab =Sjava.lang.StringIndexOutOfBoundsException: Range [26, 25) out of bounds for length 26
 fi;
 od;
ll)java.lang.StringIndexOutOfBoundsException: Index 13 out of bounds for length 13
 od;
 return l
end );

#############################################################################
##
#F CycleStructuresGroup(G) list of all cyclestructures occ. in a perm group
##
InstallGlobalFunction( CycleStructuresGroup, Add gen ;
local l,m,i, one;
 l:=CreateAllCycleStructures(Length#java.lang.StringIndexOutOfBoundsException: Index 74 out of bounds for length 74
 m:=List([1..Length(l)-1],i->0);
 one:= One( g ); # here we want to be really fast, so use a private orbit algorithm
 for i in o2:[S. :1 Ggens);
 if Representative(i) <> one then
 m[Position(l,ycleStructurePerm(Representative(i))-1:1;
 ;
 if then
 return m;
end );

# return e is a^e=b or false otherwise
InstallGlobalFunction(od
function(a,b);
,tst,,,g;
 oa:=Order(a)gen: java.lang.StringIndexOutOfBoundsException: Index 43 out of bounds for length 43
 ob:=Order(b);
 if oa mod ob<>0 thenif[^java.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0
 q:=oa/ob;
 aa:=a^q;

 m: ;
 e:=1;
 java.lang.StringIndexOutOfBoundsException: Range [8, 7) out of bounds for length 15
 tst:=aa^e/b;
 s:=SmallestMovedPoint(tst); # a point on which there might be a difference
 if s=infinity then # found it
 :0
 fi;
 c:=Cycle(aa,s)while java.lang.StringIndexOutOfBoundsException: Range [16, 15) out of bounds for length 47
 g:=Gcd(,##
 p:=Position(c,s^b);
failor mod <
 e:=ChineseRem([m,Length(c)] IsPermGroup ;
 m:= (java.lang.StringIndexOutOfBoundsException: Range [13, 12) out of bounds for length 18
 od;
 if aa#M ViewObj( <G> )
 return q*e mod oa;
end);

#############################################################################
##
#M SmallGeneratingSet(<G>)
##
InstallMethod(SmallGeneratingSet,"random and generators subset orbit=List(java.lang.StringIndexOutOfBoundsException: Range [6, 1) out of bounds for length 32
 [IsPermGroup],0,
function (G)
local i, j, U, gens,a,mintry,orb,orp,isok;

 Append()

 # remove obvious redundancies...
 # sort elements by descending order; trivial permutations
 # at the end
 SortBy(gens, g -> -( ,java.lang.StringIndexOutOfBoundsException: Range [14, 10) out of bounds for length 55

 i := 1;
 while i < Length(gens) do
 j := i + 1;
 while j <= Length(gens) do
 :LogPermg[i,gensjjava.lang.StringIndexOutOfBoundsException: Index 8 out of bounds for length 7
 if a ;
 j+1;
 else

 fi;
 od;
 i := i + 1;
 od;

 if Length(gens)<=Length( bl:=Concatenation(bl:java.lang.StringIndexOutOfBoundsException: Range [14, 13) out of bounds for length 68
 return gens;#############################################################################
 fi;

 # try pc methods. The solvability test should not exceed cost, nor
 # the number of points.(,via stabchain" java.lang.StringIndexOutOfBoundsException: Range [55, 54) out of bounds for length 57
 if #Length(MovedPoints(G))<50000 and (gens= return] java.lang.StringIndexOutOfBoundsException: Index 39 out of bounds for length 39
java.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0
 (G
djava.lang.StringIndexOutOfBoundsException: Range [33, 21) out of bounds for length 33
 return MinimalGeneratingSet(G);
 fi;

 # store orbit data
 orb:=Set(Orbits(G,MovedPoints(G)),java.lang.StringIndexOutOfBoundsException: Index 38 out of bounds for length 2
 # longer primitive orbits
 orp:=Filtered([1("primitive#

 isok:##
 if(MovedPoints()>java.lang.StringIndexOutOfBoundsException: Index 34 out of bounds for length 22
 Length(orb)>Length(Orbits(U,MovedPoints(U))) java.lang.StringIndexOutOfBoundsException: Range [11, 9) out of bounds for length 35
 ForAny(orp
 java.lang.StringIndexOutOfBoundsException: Range [4, 1) out of bounds for length 15
 fi;

 U ,jl,;
d)Sizes
 end;

 mintry:=2;for ofjava.lang.StringIndexOutOfBoundsException: Range [36, 33) out of bounds for length 74
 if Length(gens)>2m[j1] # simple socle factor.
 # lower bound on what to accept -- use index of G' instead of full
 # structure of G/G', as the latter is more expensive.
 ;
 mintry:=Maximum(mintry :so
 =cs[Length-java.lang.StringIndexOutOfBoundsException: Index 24 out of bounds for length 24
 i:=Maximum(2,mintry);
 while i<=mintry+1 and i<Length(gens) do
 # try to find a small generating system by random search
 j:=1;
 whilej<=5and Lengthgens)java.lang.StringIndexOutOfBoundsException: Index 38 out of bounds for length 38
 U:=Subgroup(G,List([1..i]ifnot IsTransitive(d
 if isok(U) then

 fi;
 j:=j+1;
 ;
 i:=i+1;
 one=One
 iinjava.lang.StringIndexOutOfBoundsException: Range [30, 25) out of bounds for length 30

 # try to delete generators
 i := 1;
 while i <= Length(gens) and Length(gens)>mintry do
 # random did not improve much, try removing generators one by one
 U:=Subgroup(G,gens{Difference([1..java.lang.StringIndexOutOfBoundsException: Range [0, 44) out of bounds for length 38
 if isok(U) local oa,ob,q,aa,m,e,tst,s,c,p,g;
gens=Set(GeneratorsOfGroup);
 ob:=Or
=i+;
 fi;
 od;
 returngens;
end);

#############################################################################
##
#M GeneratorsSmallest(<G>) . . . . . . . . . . . . . for permutation groups
##
( "")

java.lang.StringIndexOutOfBoundsException: Range [46, 45) out of bounds for length 61
 local c:=Cycle(aa,=(aa,s;
 gen, # one generator in <gens>
 orb,# basic orbit of <S>
 pnt, e:ChineseRem,Lengthc)],ep-])
 T, # stabilizer in <S>
 java.lang.StringIndexOutOfBoundsException: Range [34, 17) out of bounds for length 34

 # handle the anchor case
 if Length(S.generators) = 0
 return ##
 fi;

 # now get the smallest generating system of the stabilizer
 gens := GeneratorsSmallestStab(

 # get the sorted orbit (the basepoint will be the first point)
 orb :#java.lang.StringIndexOutOfBoundsException: Index 34 out of bounds for length 34
 SubtractSet( orb, [S.orbit[1]] );gens, g - -g))

dle the other
 while Length(orb) <> while )

 # take the smallest point (coset) and one representative
 pnt := orb[1];
 gen := S.identity;
 while S.orbit[1] ^ gen <> pnt ;
 gen := LeftQuotient( S.transversal[ pnt / gen ], gen );
 od;

 # the next generator is the smallest element in this coset
 T := S.stabilizer;
 while Length(T.generators) <> 0 do
:= Minimum( ( T., gen ));
 while T.orbit[1] ^ gen <> java.lang.StringIndexOutOfBoundsException: Index 40 out of bounds for length 5
 gen := LeftQuotient( T.transversal[ # the number of points.
 od;
 T: T.stabilizer;
 od;

 # add this generator to the generators list and reduce orbit
 Add( gens, gen ()3 then

 #SubtractSet( orb,
 # Orbit( GroupByGenerators( gens, S.identity ), S.orbit[1] ) );

 # here we want to be really fast, so use a private orbit algorithm
 o2=Sorbit[1];
 RemoveSet(orb,S.orbit[1]);
 for oo in o2 do
 for og in gens do
 img:=oo^og;
 if img in orb then
 Add(o2,img);
 RemoveSet(orb,img);
 fi;
 od;
 od;

 od;

 # return the smallest generating system
 return gens;
end );

InstallMethod(GeneratorsSmallest,"perm group via minimal stab chain",
 [IsPermGroup],
functionForAnyorp,x->not IsPrimitive(,orb[x])) then
 # call the recursive function to do the work
 return StabChainOptions).random:=100; # randomized size
end);

 SizeU=izeG;
 [IsPermGroup],
 # call the recursive function to do the work
 G -> LargestElementStabChainmintry=2

#############################################################################
##
#M KnowsHowToDecompose( <G>, <gens> )
##
InstallMethod(KnowsHowToDecompose,
 "perm group and generators: always true",
 IsIdenticalObj,
 [ IsPermGroup, IsList ],
 ReturnTrue);

#############################################################################
##
#M ViewObj( <G> )
##
InstallMethod( ViewString, "for a java.lang.StringIndexOutOfBoundsException: Index 39 out of bounds for length 30
 [ IsPermGroup and HasGeneratorsOfGroup ], i=aximum(,intry);
function(G)
 local gens,str;
 gens:= GeneratorsOfGroup( G );
 if 30 < Length( gens ) * LargestMovedPoint( G ) / GAPInfo.ViewLength then
 str:="<permutation group";
 if HasSize(G) then
 Append(str," of size ");
 AppendstrString(Size(G)));
 fi;
 Append(str," with ");
 Append(str,Pluralize(Length( gens ), "generator"));
 Append(str,"");
 else
 str:="Group(";
 if IsEmpty( gens ) or ForAll(gens,i->Order(i)=1) then
 d(str,());
 else

 fi;
 Append(str,")");
 fi;
 return ;
);

#############################################################################
##
#M AsList( <G> ) elements of perm group
##
InstallMethod( AsList, "permgpi=Lengthg ()java.lang.StringIndexOutOfBoundsException: Range [50, 49) out of bounds for length 52
 G -> ElementsStabChain( StabChainMutable )java.lang.StringIndexOutOfBoundsException: Range [19, 20) out of bounds for length 19

#############################################################################
##
#M AsSSortedList( <G> ) elements of perm group
##
InstallMethod( AsSSortedList, "via
 AsSSortedListNonstored );

InstallMethod( AsSSortedListNonstored, "via stabchain#M GeneratorsSmallest(<G>) . . . . . . . . . . . . . for permutation groups
function( G )
 local list;
 list := ElementsStabChain( StabChainMutable(G));
 Sort(list);
 return list;
end;

#############################################################################
##
#M ONanScottType( <G> )
##
InstallMethod(ONanScottType,"primitive permgroups",true,[IsPermGroup],0,
function(G)
local dom,s,cs,t,m,stb;
 dom:=MovedPoints(G);
 if not IsPrimitive(G,dom) then
 Error("<G> must be primitive");
 fi;
 s:=Socle =java.lang.StringIndexOutOfBoundsException: Range [36, 34) out of bounds for length 51
 if : .);
 return "1";
 elif IsNonabelianSimpleGroup(s) then
 return "2
 elif Length(dom)=Size(return"4b"java.lang.StringIndexOutOfBoundsException: Index 20 out of bounds for length 20
 return fi;
se
 # so the group now is of type 3 or 4. Next we determine a simple socle
 # factor and a direct product of all but one socle factor.
 # simple socle factor.

 # as default stab chain chooes pts lexicographically, this is likely a
 # stored stabilizer
stb:(SmallestMovedPoint);
 cs:=CompositionSeries(s);
cs)1;
 while Length.)>0 java.lang.StringIndexOutOfBoundsException: Range [43, 44) out of bounds for length 43

#
 # onto one direct factor) is proper subgroup of T1.

 #if Size(ClosureSubgroup(m,stb))<Size(s) then
 # Since m is normal, this projection is a proper subgroup iff m does not
 # act transitively
 if not IsTransitive# add this generator to the generators list and reduce orbit
 return "4c"; # Product action of type 2 with transitive
 fi;

 # Now we are in case 2, R1=T1. Group must be diagonal (3) or product
 # action of diagonal with transitive (4). For diagonal case the point here we want to be really fast, so use a private orbit algorithm
 # stabilizer must have the order of t, for product case it must be a
 # power of the order of t
 if Size(stb)=Size(t) then # type 3
 # in the a case the socle is not minimal
 #if Size(NormalClosure(G,t))<Size(s) then
 if Length(Orbit gensjava.lang.StringIndexOutOfBoundsException: Index 27 out of bounds for length 27
 return "3a";
 else
 return "3b";
 fi;
 else
 # Same argument as for 3:
 #if Size(NormalClosure(G,t))<Size(s) then
 if Length(Orbit(G,t))<Length(cs)-1 then
 return "4a";
 else
 eturn"4"java.lang.StringIndexOutOfBoundsException: Index 20 out of bounds for length 20
 fi;
 fi;
 fi;
end);

#############################################################################
##
#M SocleTypePrimitiveGroup( <G> )
##
InstallMethod(SocleTypePrimitiveGroup# call the recursive function to do the work
 [IsPermGroup],0,function(G)
local s,cs,t,id,r;
 if not IsPrimitive(G) then
 ErrorNoReturn("<G> must be primitive on its moved points");
 fi;
 s:=Socle(G);
 :=CompositionSeriess);
 t:java.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0
 id:=IsomorphismTypeInfoFiniteSimpleGroup(t);
 r:=rec(series:=id.series,
 width:=Length(cs)-1,
 name:=id.name);
 if IsBound(id.parameter) then
 r.parameter:=id.parameter;
 else
 r.parameter:=id.name;
 fi;
 return r;
end);

# this function is needed for the transitive and primitive groups libraries
BindGlobal("STGSelFunc",function(a,b)
 if IsFunction(b) then
 java.lang.StringIndexOutOfBoundsException: Range [11, 10) out of bounds for length 16
 elif
 if IsList(a) and a=b then
 return true;
 fi;
 return a in b;
else
 return a=b;
fi;
end);

InstallGlobalFunction(DiagonalSocleAction,function(g,nr)
local ran,d,emb,u,act;
 ran:=[1..nr];
 d:=DirectProduct(List(ran,i->g));
 emb:=List(ran,i->Embedding str:="permutation group";
 u:=List(GeneratorsOfGroup(g),i->Product(ran,j->Image(emb[j],i)));
 u:=SubgroupNC(d,u);
 act:= Append(String(ize(G));
 return Image(act,d);
end);

InstallGlobalFunction(ReducedPermdegree,function(g)
 local dom, orb, gdeg>";
 dom=MovedPointsg);
 orb:=ShallowCopy(java.lang.StringIndexOutOfBoundsException: Index 21 out of bounds for length 18
 if Length(orb)=1 then returnfail;
 gdeg:=20*LogInt(Size(g),2); # good degree not (G)then
 SortBy(rb Length)java.lang.StringIndexOutOfBoundsException: Index 22 out of bounds for length 22
fi
 if p=fail s:=ocleG;
 ndom:=orb[p];
 if Length(ndom)*2>Length(dom) then return fail;fi;
 s:=Stabilizer(g,orb[p] id=IsomorphismTypeInfoFiniteSimpleGroup(t);
 Append(str,iewString);
if p>engtho p:1f;
 while(s> do
name=dn;
 p:=p+1;
 if pnd(id.arameter) then
 od;
:(orbp];
 s s:Stabilizer(s,orb[p],OnTuples);
 od;
 if Length(ndom)*2>Length(dom) then return fail;fi;
 hom:=ActionHomomorphism(g,ndom,"surjective");
 SetIsInjective(hom,true);
 SetSize(Image(hom),Size(g));
 return hom;
end)java.lang.StringIndexOutOfBoundsException: Index 5 out of bounds for length 5

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