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(CanEasilyTestMembership,IsPermGroup);

#############################################################################
##
#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 72 out of bounds for length 72

#M CanComputeSizeAnySubgroup
InstallTrueMethod(CanComputeSizeAnySubgroup,IsPermGroup);

#############################################################################
##
#M AsSubgroup( <G>, ) . . . . . . . . . . . . with stab chain transfer
##
InstallMethod( base, # the
 size,# the size of the group generated by <inds>
 functionorbs #java.lang.StringIndexOutOfBoundsException: Index 32 out of bounds for length 32
 local S;
 # test if the parent is already alrightone,# identity of `P'
 if HasParent(U) and IsIdenticalObj(Parent(U),G) then

 ;

 if , # one generator from <gens>
 return fail;
 fi;
 S:= c, # other point in orbit
 UseIsomorphismRelation( U, S )java.lang.StringIndexOutOfBoundsException: Index 35 out of bounds for length 35
 UseSubsetRelation( U, S );
 if HasStabChainMutable(U)then
 = [];
 fisize := 1;
 return S;
end );

#############################################################################
##
#M CanEasilyComputeWithIndependentGensAbelianGroup( <permgroup> )
##
InstallTrueMethod(CanEasilyComputeWithIndependentGensAbelianGroup,


#############################################################################
##
#F IndependentGeneratorsAbelianPPermGroup( , ) . . . nice generators
##
stallGlobalFunction( IndependentGeneratorsAbelianPPermGroup,
 function ( P, p )
 ,
 powsjava.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0
 base, # the base of the vectorspace
 for inexp,exp-1.] do
 over the remaining generators
 gens2 :=]java.lang.StringIndexOutOfBoundsException: Index 20 out of bounds for length 20
 : gens[]java.lang.StringIndexOutOfBoundsException: Index 25 out of bounds for length 25
 ;
 gens2,
 exp
 g, # one generator from <gens>
 # its power
 b, # basepoint
 c, Add powsg);
 i ,k loop

 # initialize the list of independent generators = size * java.lang.StringIndexOutOfBoundsException: Index 35 out of bounds for length 35
 inds := [];
 pows := [];
 base :### Copyright of GAP belongs to its developers, whose names are too numerous;
 size := 1;
 orbsjava.lang.StringIndexOutOfBoundsException: Index 8 out of bounds for length 2
 ## CanEasilyTestMembership

 #############################################################################
 java.lang.StringIndexOutOfBoundsException: Range [0, 8) out of bounds for length 2
 ## imp

 # loop over the exponents
 exp := Maximum( List( gens, g - odjava.lang.StringIndexOutOfBoundsException: Index 23 out of bounds for length 23
 for Add(java.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0

(java.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0
 gens2= [];
 in1.gens) java.lang.StringIndexOutOfBoundsException: Range [39, 40) out of bounds for length 39
 g := gens[j];
 ##

 IsAbelian,
 notnot
java.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0
java.lang.StringIndexOutOfBoundsException: Index 24 out of bounds for length 14
 returnfail;
 h :p #java.lang.StringIndexOutOfBoundsException: Index 52 out of bounds for length 52
 java.lang.StringIndexOutOfBoundsException: Index 16 out of bounds for length 15

 # if this is linear independent, add it to the generators
 ;
 ( g;
 Add( powsAdd( p< 1

 java.lang.StringIndexOutOfBoundsException: Range [13, 12) out of bounds for length 19
h;
 if not IsBound( orbs

 []: [ Append(inds,
 fi;
 for c in
 for k odjava.lang.StringIndexOutOfBoundsException: Index 7 out of bounds for length 7
 (orbsb,c ^ (h^k ;
 trns[c^(h^k)] java.lang.StringIndexOutOfBoundsException: Index 2 out of bounds for length 2
 od;


 java.lang.StringIndexOutOfBoundsException: Index 16 out of bounds for length 16
g)



 orb:= [d;

 # prepare for the next round
s:= gens2;
 newd := java.lang.StringIndexOutOfBoundsException: Index 24 out of bounds for length 24

 ge :=(

 # return the independent generators
 inds (gens , java.lang.StringIndexOutOfBoundsException: Index 65 out of bounds for length 65
end fornewmg]then

#############################################################################
##
#M IndependentGeneratorsOfAbelianGroup( <G> ) . . . . . . . nice generators
##
while IsOne()
 [ IsPermGroupjava.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0
 function
 m(]]

 # ofgroupsize
 gens, # generators of -Sylowsubgrouplargest> java.lang.StringIndexOutOfBoundsException: Range [57, 56) out of bounds for length 60
g, onegenerator
 # its order

 # loop over all primes
 : ;
 one:= One( G );
 for p in Union( List fst = [b ]
 g
 ifforc ShallowCopy(rbs inorb do

 # compute the generators for the Sylow subgroup for gensdo
 gens := [];
 for g in GeneratorsOfGroup( G
 o := Order(g);
while p0 doo: / p;;
 = ;
 if not IsOne(g) then java.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0
java.lang.StringIndexOutOfBoundsException: Index 31 out of bounds for length 11

 # append the independent generators for the Sylow subgroupjava.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0
 Append( inds,
 java.lang.StringIndexOutOfBoundsException: Index 27 out of bounds for length 11
 GroupByGenerators (

 ;
 od;

 # Sort the independent generators by increasing order(orbs );
 SortBy

 # return the independent generators ;
 g,
end ) o # its order

#############################################################################
##
#F OrbitPerms( <gens>, <d> ) . . . . . . . . . . . . . orbit of permutations
##
InstallGlobalFunction( in FactorsOrder() )do
 orb, ) then

 # get the largest point <max> moved by the group <G>
 m <minthen

 # handle fixpoints
 if gens :[];
 return [ d ];
 in ( )do

 # start with the singleton orbit o p od
 orb := [ : g;
 new IsOneg then ( ,g; java.lang.StringIndexOutOfBoundsException: Index 54 out of bounds for length 54
 new[dSortBy(inds,Order)java.lang.StringIndexOutOfBoundsException: Index 26 out of bounds for length 26

 # loop over all points found
 for pnt java.lang.StringIndexOutOfBoundsException: Range [0, 2) out of bounds for length 0

 # apply all generators <gen>
 for gen inmax =java.lang.StringIndexOutOfBoundsException: Range [10, 0) out of bounds for length 0
 imgjava.lang.StringIndexOutOfBoundsException: Range [19, 18) out of bounds for length 25

 # add the image <img> to the orbit if it is new
 if new[img] then

 new[imgjava.lang.StringIndexOutOfBoundsException: Range [12, 3) out of bounds for length 32
 fi;

 odlocal java.lang.StringIndexOutOfBoundsException: Range [24, 23) out of bounds for length 28
 #####################################################################java.lang.StringIndexOutOfBoundsException: Index 15 out of bounds for length 2
end );

#############################################################################
##
#F OrbitsPerms( <gens>, <D> ) . . . . . . . . . . . orbits of permutations
##
InstallGlobalFunction(
 # get the largest point <max> moved by the group <G>

 java.lang.StringIndexOutOfBoundsException: Index 60 out of bounds for length 60
 max :orbsod;
 # repeat until the domain is exhausted
 new =java.lang.StringIndexOutOfBoundsException: Range [27, 24) out of bounds for length 47
 orbs

 # repeat until the domain is exhausted
 fst
do


 :[ fstjava.lang.StringIndexOutOfBoundsException: Index 27 out of bounds for length 27
 #############################################################################
 dom[ for

 # loop over all points found
 for pnt in

 # apply all generators <gen>
 for
 img : ^gen;


 java.lang.StringIndexOutOfBoundsException: Index 15 out of bounds for length 15
 Add( orb
 [ =java.lang.StringIndexOutOfBoundsException: Index 42 out of bounds for length 42
 dom[img


 od;

 od;

 # add the orbit to the list of orbits and take next point
"a permutation group"java.lang.StringIndexOutOfBoundsException: Index 30 out of bounds for length 30
 >LargestMovedPoint java.lang.StringIndexOutOfBoundsException: Range [46, 45) out of bounds for length 55

 od;

 # add the remaining points of <D>, they are fixed
pnt D
 if pnt>max then
 Add( java.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0
 ;
 od;

 return orbsjava.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0
end< ) ..

#############################################################################
##
#M SmallestMovedPoint( <C> ) . . . . . . . for a collection of permutations
##
BindGlobal -java.lang.StringIndexOutOfBoundsException: Range [22, 20) out of bounds for length 49
 java.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0

 min := infinityjava.lang.StringIndexOutOfBoundsException: Index 24 out of bounds for length 2
 for [,
 NrMovedPoints( )

 if m < min
 java.lang.StringIndexOutOfBoundsException: Range [5, 3) out of bounds for length 17
 fi;
 fi;
 od;
 return
;

InstallMethod;
 ;
 truee )
 #M \=( <G>,
 SmallestMovedPointPerms );

java.lang.StringIndexOutOfBoundsException: Range [9, 8) out of bounds for length 9
 notjava.lang.StringIndexOutOfBoundsException: Index 9 out of bounds for length 9

 [
 SmallestMovedPointPerms##

#############################################################################
##
#M LargestMovedPoint( <C> ) . . . . . . . for a collection of permutations
##
BindGlobal( "java.lang.StringIndexOutOfBoundsException: Index 26 out of bounds for length 5
 local max

 ;
 forG= ( do
 not (
 m := LargestMovedPointPerm( gen
java.lang.StringIndexOutOfBoundsException: Index 28 out of bounds for length 28
 max i java.lang.StringIndexOutOfBoundsException: Index 30 out of bounds for length 30
 ;
 fi;
 od;
 #
end );

InstallMethod( java.lang.StringIndexOutOfBoundsException: Index 27 out of bounds for length 5
 java.lang.StringIndexOutOfBoundsException: Index 1 out of bounds for length 0
 true,
java.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0
 LargestMovedPointPerms );

InstallMethod(#M Enumerator(
 "for = java.lang.StringIndexOutOfBoundsException: Index 11 out of bounds for length 2
java.lang.StringIndexOutOfBoundsException: Range [9, 8) out of bounds for length 9
 [ IsEmptyjava.lang.StringIndexOutOfBoundsException: Index 7 out of bounds for length 7
 LargestMovedPointPerms;

#############################################################################
##
#F MovedPoints( <C> ) . . . . . . . . . . for a collection of permutations
##
InstallGlobalFunction( then
 local,,genpntjava.lang.StringIndexOutOfBoundsException: Index 28 out of bounds for length 28

 mov := [ ];
 for gen infi =QuoIntpos, )
 if gen <java.lang.StringIndexOutOfBoundsException: Index 16 out of bounds for length 0
ovedPoint gen).java.lang.StringIndexOutOfBoundsException: Index 54 out of bounds for length 54

 if"for ;
 ( "NumberElemen"java.lang.StringIndexOutOfBoundsException: Index 38 out of bounds for length 38
 fi notnot(elm
 od;
 fi;
 od;
 java.lang.StringIndexOutOfBoundsException: Index 6 out of bounds for length 2
end);

InstallMethodS .;
 [ Length g java.lang.StringIndexOutOfBoundsException: Range [9, 8) out of bounds for length 30
 MovedPointsPerms)

#############################################################################
##
 MovedPointsPerms java.lang.StringIndexOutOfBoundsException: Range [28, 22) out of bounds for length 30

java.lang.StringIndexOutOfBoundsException: Index 13 out of bounds for length 0
function(p##
 return MovedPointsPerms([p]);
end);

#############################################################################
##
#M NrMovedPoints( <C> ) . . . . . . . . . for a collection of permutations

###############################################################################
 local mov, sma,

 mov := 0java.lang.StringIndexOutOfBoundsException: Range [30, 22) out of bounds for length 30
 java.lang.StringIndexOutOfBoundsException: Range [8, 7) out of bounds for length 35
 = then
 return 0;
 fi;
 fori=ist(,ength
 for:#M OrbitsMovedPo( G
 ( " a ,[IsPermGroupjava.lang.StringIndexOutOfBoundsException: Index 75 out of bounds for length 75
 mov:= mov +local o,;
java.lang.StringIndexOutOfBoundsException: Index 18 out of bounds for length 16
 fi;
 :List)
 od;
 java.lang.StringIndexOutOfBoundsException: Range [11, 10) out of bounds for length 15
end );

InstallMethod( NrMovedPoints,
: java.lang.StringIndexOutOfBoundsException: Index 15 out of bounds for length 15
 [ IsPermCollectionjava.lang.StringIndexOutOfBoundsException: Range [0, 39) out of bounds for length 11
 NrMovedPointsPerms java.lang.StringIndexOutOfBoundsException: Index 23 out of bounds for length 0

##
 "anemptyjava.lang.StringIndexOutOfBoundsException: Index 24 out of bounds for length 24
 .:]java.lang.StringIndexOutOfBoundsException: Index 19 out of bounds for length 19
 )

#############################################################################
##
#M LargestMovedPoint( <G> ) . . . . . . . . . . . . for a permutation group
##
stMovedPoint
 ";

 [ IsPermGroupfi


#############################################################################
##
#M SmallestMovedPoint( <G> ) . . . . . . . . . . . . for a permutation group
##
InstallMethodifIsEqualSet( G , GeneratorsOfGroup(H) ) java.lang.StringIndexOutOfBoundsException: Index 70 out of bounds for length 70
 "for permutation group"java.lang.StringIndexOutOfBoundsException: Index 30 out of bounds for length 30
 ifHasStabChainMutable()or HasStabChainImmutable(H) then
 [ IsPermGroup ], 0,
 G -> SmallestMovedPoint( GeneratorsOfGroup( G ) oup(),gen->gen in H);

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

#############################################################################
##
#M NrMovedPoints( <G> ) . . . . . . . . . . . . . . for a permutation group
##
InstallMethod
 for group,
 true,
 [ IsPermGroup ], 0,
 G -> NrMovedPointstrue,

#############################################################################
##
#M OrbitsMovedPoints( <G> )
##
InstallMethod( OrbitsMovedPoints, "for a permutation group", [IsPermGroup],
unction(G)
local o,i;
 o:=Orbits(G,MovedPoints(G)); );
 o:=Listjava.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0
 List(o,IsRange); # save memory if long orbits
 Sort(o);
 MakeImmutableo)java.lang.StringIndexOutOfBoundsException: Index 19 out of bounds for length 19
 for i
 (,true)
 od;
 SetIsSSortedList(o,true);
 java.lang.StringIndexOutOfBoundsException: Range [9, 8) out of bounds for length 11
java.lang.StringIndexOutOfBoundsException: Range [4, 3) out of bounds for length 30

#############################################################################
##
#M \=( <G>, <H> ) . . . . . . . . . . . test if two perm groups are equal
##
java.lang.StringIndexOutOfBoundsException: Range [14, 13) out of bounds for length 79
( G,HiG)java.lang.StringIndexOutOfBoundsException: Index 31 out of bounds for length 31
 # avoid new stabilizer chains if they are not computed yet
) : .java.lang.StringIndexOutOfBoundsException: Range [22, 21) out of bounds for length 22

 if not (HasGeneratorsOfGroup(G) and java.lang.StringIndexOutOfBoundsException: Index 55 out of bounds for length 19
 len ;
 fieelm=

 if java.lang.StringIndexOutOfBoundsException: Index 18 out of bounds for length 0
 or S := S.;
 return false;
 fi;

 IsEqualSet ajava.lang.StringIndexOutOfBoundsException: Range [27, 22) out of bounds for length 49
 return true;
 >, java.lang.StringIndexOutOfBoundsException: Range [35, 34) out of bounds for length 61

 # try to use one existing stab chainBindGlobal( "N"java.lang.StringIndexOutOfBoundsException: Index 38 out of bounds for length 38
 if HasStabChainMutable(H)en
java.lang.StringIndexOutOfBoundsException: Range [13, 12) out of bounds for length 18
 fi;
 return Size(java.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0
end);

#############################################################################
##
#M BaseOfGroup( <G> )
##

roup,img= .orbitP ,# the
 true,
 [
 G -> BaseStabChain# new generator list

#############################################################################
##
#M Size( <G> ) . . . . . . . . . . . . . . . . . . size of permutation group
##
 F(, java.lang.StringIndexOutOfBoundsException: Range [45, 42) out of bounds for length 50
"for group,
 true,
 [ IsPermGroupreturnjava.lang.StringIndexOutOfBoundsException: Index 33 out of bounds for length 33
 G->SizeStabChain (G)jj in) then

#############################################################################
##
#M Enumerator( <G> ) . . . . . . . . . . . . enumerator of permutation group
##
BindGlobal( "ElementNumber_PermGroup"end)
 function( enum, pos )
 local elm, S java.lang.StringIndexOutOfBoundsException: Range [40, 39) out of bounds for length 70

 S := enum!.stabChain;
 elm size:ptionsimit;java.lang.StringIndexOutOfBoundsException: Index 34 out of bounds for length 34
 n: pos;
 :=os - ;
 while Length( p:=]; :=
= i))java.lang.StringIndexOutOfBoundsException: Index 24 out of bounds for length 24
=LeftQuotient
 ( S, S.java.lang.StringIndexOutOfBoundsException: Index 33 out of bounds for length 31

 : :(G(ase=);
 od;

 ifnot(java.lang.StringIndexOutOfBoundsException: Index 44 out of bounds for length 44
NumberElement
 fi;
 return elm;java.lang.StringIndexOutOfBoundsException: Index 13 out of bounds for length 0
end );

BindGlobalPrint
 function( # if inpar then
 local# java.lang.StringIndexOutOfBoundsException: Range [59, 58) out of bounds for length 74


 Length(S#java.lang.StringIndexOutOfBoundsException: Index 32 out of bounds for length 32
 fi;

 pos := 1;
 = 1java.lang.StringIndexOutOfBoundsException: Index 13 out of bounds for length 13
 S := enum!.UseSubsetRelat C, G java.lang.StringIndexOutOfBoundsException: Index 37 out of bounds for length 37
# ;
imgjava.lang.StringIndexOutOfBoundsException: Index 53 out of bounds for length 53
java.lang.StringIndexOutOfBoundsException: Range [9, 8) out of bounds for length 11

:java.lang.StringIndexOutOfBoundsException: Range [39, 8) out of bounds for length 11

 val := val * options.base := Base;
 elm ifjava.lang.StringIndexOutOfBoundsException: Range [33, 32) out of bounds for length 68
 bilizer;
 else
 return fail;
 fi;
 od;
 if elm <> Sjava.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0

 ;
 returnif##
java.lang.StringIndexOutOfBoundsException: Range [17, 6) out of bounds for length 6

InstallMethod( java.lang.StringIndexOutOfBoundsException: Index 25 out of bounds for length 20
#
 [ java.lang.StringIndexOutOfBoundsException: Range [16, 15) out of bounds for length 65
function(G)
local : java.lang.StringIndexOutOfBoundsException: Index 25 out of bounds for length 7

 ;
 p:=
 while return rnd
 ejava.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0
ength
 ijava.lang.StringIndexOutOfBoundsException: Index 2 out of bounds for length 2
 Add(p,e,IsEmpty, java.lang.StringIndexOutOfBoundsException: Index 47 out of bounds for length 47
 ScInstallMethodjava.lang.StringIndexOutOfBoundsException: Range [30, 28) out of bounds for length 55
 od;
 Info =java.lang.StringIndexOutOfBoundsException: Range [31, 29) out of bounds for length 35

 S:=java.lang.StringIndexOutOfBoundsException: Index 15 out of bounds for length 6
 e:=#M ClosureGroup( (G,
 java.lang.StringIndexOutOfBoundsException: Range [0, 8) out of bounds for length 2
 "(G java.lang.StringIndexOutOfBoundsException: Range [56, 54) out of bounds for length 56
Length >java.lang.StringIndexOutOfBoundsException: Range [52, 51) out of bounds for length 71
 java.lang.StringIndexOutOfBoundsException: Range [27, 21) out of bounds for length 43

 end,

 = )
<0do
 S.ol:java.lang.StringIndexOutOfBoundsException: Index 8 out of bounds for length 0
 S.orbitposif java.lang.StringIndexOutOfBoundsException: Range [14, 13) out of bounds for length 25
 for i in [1..Maximum(S.orbit ForAlljava.lang.StringIndexOutOfBoundsException: Index 13 out of bounds for length 13
 (S.,)java.lang.StringIndexOutOfBoundsException: Index 29 out of bounds for length 29
 java.lang.StringIndexOutOfBoundsException: Range [10, 8) out of bounds for length 21
 ,:(java.lang.StringIndexOutOfBoundsException: Range [28, 27) out of bounds for length 70
 fi;
 od;
 S:=java.lang.StringIndexOutOfBoundsException: Range [12, 10) out of bounds for length 74
 od;
 ;
end);

#############################################################################
##
#M Iterator( <G> ) . . . . . . . . . . . . . . iterator of permutation group
##
InstallMethod java.lang.StringIndexOutOfBoundsException: Range [45, 44) out of bounds for length 66
 " gensG := GeneratorsOf);
 java.lang.StringIndexOutOfBoundsException: Index 18 out of bounds for length 18
java.lang.StringIndexOutOfBoundsException: Range [11, 8) out of bounds for length 11
java.lang.StringIndexOutOfBoundsException: Range [11, 10) out of bounds for length 30
end);

InstallMethod injava.lang.StringIndexOutOfBoundsException: Range [34, 33) out of bounds for length 61
 "for a permutation # if DefaultStabChainOptions.tryPcgs
 [ IsPermGroup ],
# if inpar temp SubgroupNCjava.lang.StringIndexOutOfBoundsException: Index 32 out of bounds for length 32
 return IteratorStabChain# C := GroupByGenerators
end);

#############################################################################
##
#M Random( <G> ) . . . . . . . . . . . . . . . . . . . . . . random element
##
# Add( gensN)
 "
 [ # make the base
 function: java.lang.StringIndexOutOfBoundsException: Range [29, 22) out of bounds for length 37
 ,;

 # go down the stabchain and multiply random representatives
 S := ifgensN :( java.lang.StringIndexOutOfBoundsException: Range [45, 43) out of bounds for length 51
 rnd identityjava.lang.StringIndexOutOfBoundsException: Index 22 out of bounds for length 22
 while Length( Sthejava.lang.StringIndexOutOfBoundsException: Range [44, 43) out of bounds for length 69
 rnd := LeftQuotient java.lang.StringIndexOutOfBoundsException: Index 38 out of bounds for length 33
 java.lang.StringIndexOutOfBoundsException: Index 11 out of bounds for length 11
 S := S.stabilizer
 od;

 # return the random element
 return rndnotorbits java.lang.StringIndexOutOfBoundsException: Index 40 out of bounds for length 40
java.lang.StringIndexOutOfBoundsException: Index 24 out of bounds for length 6

#############################################################################
##
#M <g> in <G> . . . . . . . . . . . . . . . . . . . . . . . membership test
##
InstallMethod
 ":java.lang.StringIndexOutOfBoundsException: Range [43, 36) out of bounds for length 46
 true,
 [ IsPerm, java.lang.StringIndexOutOfBoundsException: Index 21 out of bounds for length 0
java.lang.StringIndexOutOfBoundsException: Index 72 out of bounds for length 20
 if g = Onejava.lang.StringIndexOutOfBoundsException: Range [26, 24) out of bounds for length 32
 java.lang.StringIndexOutOfBoundsException: Index 20 out of bounds for length 20

 InstallMethod( ClosureGroup, "permgroup, element",true,
 return SiftedPermutation
 ()java.lang.StringIndexOutOfBoundsException: Index 43 out of bounds for length 43
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.
"oClosurePrmGpf G, gens, )
 local C, # closure of < <G>, <obj> >, result IsPermGroup ],0java.lang.StringIndexOutOfBoundsException: Index 39 out of bounds for length 39
 ,inpar, # parent of the closure
 g, # an element of gens
 o, # order
 ,# new generator list
 chain; # the stabilizer chain created

 options:=ShallowCopy(java.lang.StringIndexOutOfBoundsException: Index 28 out of bounds for length 25
 # if all generators are in <G>, <G> is the closure
 gens := java.lang.StringIndexOutOfBoundsException: Index 18 out of bounds for length 0
 if IsEmpty( gens ) then
 return G;
 fi;

 # is the closure normalizing?
 if Length(gens)=1 and
 ForAll(gens,i. correctthen
 :ens1]
 if Size(Gjava.lang.StringIndexOutOfBoundsException: Range [11, 10) out of bounds for length 11
 o:=SetStabChainMutablejava.lang.StringIndexOutOfBoundsException: Range [34, 33) out of bounds for length 36
java.lang.StringIndexOutOfBoundsException: Index 13 out of bounds for length 13
 else
 o:=First(Difference(##java.lang.StringIndexOutOfBoundsException: Range [21, 20) out of bounds for length 39
 java.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0
 fi
 options.size ;
 end );
 fi;

 if not IsSubset(G,U) then
 java.lang.StringIndexOutOfBoundsException: Range [52, 51) out of bounds for length 58
 = P, )
 while not inpar and
 P := Parent# gensG := GeneratorsOfGroup( G );
 inpar :java.lang.StringIndexOutOfBoundsException: Index 43 out of bounds for length 43
 od;
 if inpar then
 := GroupByGenerators( , ( java.lang.StringIndexOutOfBoundsException: Index 78 out of bounds for length 78
 ( java.lang.StringIndexOutOfBoundsException: Range [26, 24) out of bounds for length 59
 =java.lang.StringIndexOutOfBoundsException: Range [50, 49) out of bounds for length 57
 fi;

 C>java.lang.StringIndexOutOfBoundsException: Range [56, 48) out of bounds for length 61

#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;

 # make the base of G compatible with options.base java.lang.StringIndexOutOfBoundsException: Range [15, 13) out of bounds for length 41
 # java.lang.StringIndexOutOfBoundsException: Range [0, 19) out of bounds for length 14
 if IsBoundfori java.lang.StringIndexOutOfBoundsException: Range [20, 19) out of bounds for length 30
 ChangeStabChain( if not cnj in N then
IsBound(.java.lang.StringIndexOutOfBoundsException: Range [41, 40) out of bounds for length 65
java.lang.StringIndexOutOfBoundsException: Index 26 out of bounds for length 7

 if java.lang.StringIndexOutOfBoundsException: Range [12, 24) out of bounds for length 15
 gens java.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0
 options := ShallowCopy( C := ClosureGroup( C,recjava.lang.StringIndexOutOfBoundsException: Range [56, 53) out of bounds for length 59
 b :: java.lang.StringIndexOutOfBoundsException: Range [32, 29) out of bounds for length 35
 for g in gens do
 java.lang.StringIndexOutOfBoundsException: Range [40, 39) out of bounds for length 68
 StabChainStrong( chain, [ = :=java.lang.StringIndexOutOfBoundsException: Range [38, 37) out of bounds for length 46
 # force closure test
 od;
 else(.ndom:r;
 o:=ValueOption("knownClosureSize");
 IsIntthenl:=;
 chain
 fi;

 java.lang.StringIndexOutOfBoundsException: Range [11, 10) out of bounds for length 34
 java.lang.StringIndexOutOfBoundsException: Range [23, 22) out of bounds for length 77
 if inpar then C := GroupStabChain( P, fi
 else C chain : rchain [;
 else
 if inpar thenjava.lang.StringIndexOutOfBoundsException: Index 8 out of bounds for length 8
lse C: ,() fi;
 =[,,,];
 fi;
 SetStabChainOptions( C, rec( ,k+2)4

 UseSubsetRelation( C, fi
 return C;
end)java.lang.StringIndexOutOfBoundsException: Index 6 out of bounds for length 6

therMethod;
 true,
[ andjava.lang.StringIndexOutOfBoundsException: Range [47, 46) out of bounds for length 62
 java.lang.StringIndexOutOfBoundsException: Range [19, 18) out of bounds for length 20

InstallOtherMethod resultchain.java.lang.StringIndexOutOfBoundsException: Range [36, 35) out of bounds for length 64
[ :java.lang.StringIndexOutOfBoundsException: Range [48, 39) out of bounds for length 56
ReturnFirst);

InstallMethod( ClosureGroup, " java.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0
 [ #M DerivedSubgroupifresult<c.java.lang.StringIndexOutOfBoundsException: Range [33, 31) out of bounds for length 37
G java.lang.StringIndexOutOfBoundsException: Index 16 out of bounds for length 16
 (
 ,# random subproducts of generators

java.lang.StringIndexOutOfBoundsException: Range [28, 27) out of bounds for length 56
 [ IsPermGroup list, java.lang.StringIndexOutOfBoundsException: Range [31, 30) out of bounds for length 45
function( G, gens )
 return DoClosurePrmGp N;
end );

java.lang.StringIndexOutOfBoundsException: Range [20, 18) out of bounds for length 69
[ D :java.lang.StringIndexOutOfBoundsException: Range [31, 24) out of bounds for length 30
function( G, g, options )
( , java.lang.StringIndexOutOfBoundsException: Range [18, 17) out of bounds for length 27
end );

InstallOtherMethod( ClosureGroup, "permgroup, permgroup, options")
 [ IsPermGroup, IsPermGroup, IsRecord ]##
function( G
 return DoClosurePrmGp( G, GeneratorsOfGroup( java.lang.StringIndexOutOfBoundsException: Range [29, 28) out of bounds for length 37
end ( H := GroupByGenerators( O() , G

java.lang.StringIndexOutOfBoundsException: Range [19, 18) out of bounds for length 64
IsPermGroup, H));
ReturnFirst);

#############################################################################
##
#M NormalClosureOp( <G>, ) . . . . . . . . . . . . . . . . in perm group
##
BindGlobal("java.lang.StringIndexOutOfBoundsException: Range [12, 36) out of bounds for length 17
 local f
 chain,
 rchain
 options, # options record for stabilizer chain construction
 ,# generators of the group <G>
 java.lang.StringIndexOutOfBoundsException: Range [9, 8) out of bounds for length 15

 genN, # one generator of the group <N>
 cnj, # conjugated of a generator of 
 java.lang.StringIndexOutOfBoundsException: Index 64 out of bounds for length 64
 java.lang.StringIndexOutOfBoundsException: Range [47, 46) out of bounds for length 56

 odjava.lang.StringIndexOutOfBoundsException: Index 12 out of bounds for length 12
 gensG := GeneratorsOfGroup( G );
 one:= One( G );

 # make a copy of the group to be closed
 N:= java.lang.StringIndexOutOfBoundsException: Range [11, 10) out of bounds for length 51
 UseIsomorphismRelation(
RelationUStabChainOption)randomDjava.lang.StringIndexOutOfBoundsException: Range [69, 61) out of bounds for length 69
 SetStabChainMutable( N, D := DoNormalClosurePermGroup
 :java.lang.StringIndexOutOfBoundsException: Index 26 out of bounds for length 0
 if IsBound( optionsjava.lang.StringIndexOutOfBoundsException: Index 27 out of bounds for length 0
 else
 options.temp := true;

 # make list of conjugates to be added to N
## This is a most interesting function. It tests whether a permutation
 gensN := [ ];
 for i in [;
 =java.lang.StringIndexOutOfBoundsException: Range [40, 39) out of bounds for length 59
 cnj ## Finding Composition Factors of Permn\eq10^,
 (N),java.lang.StringIndexOutOfBoundsException: Range [41, 28) out of bounds for length 69
 if not local , domainof<
 Add( gensN, java.lang.StringIndexOutOfBoundsException: Range [12, 1) out of bounds for length 39
 fi;
 # force closure test

 N := java.lang.StringIndexOutOfBoundsException: Index 19 out of bounds for length 0
 fi;
 IsTrivial;

alljava.lang.StringIndexOutOfBoundsException: Range [39, 35) out of bounds for length 77
 java.lang.StringIndexOutOfBoundsException: Range [7, 6) out of bounds for length 7
gensN
 for genN in java.lang.StringIndexOutOfBoundsException: Range [9, 8) out of bounds for length 15

 # loop over the generators of G java.lang.StringIndexOutOfBoundsException: Range [35, 33) out of bounds for length 55
 notcomm C

 Add( list, comm );
 cnj := genN ^
 if not cnj java.lang.StringIndexOutOfBoundsException: Range [4, 10) out of bounds for length 7
 N := ClosureGroupelifSize(DerivedSubgroup ))<SizeG ) then
 Add( gensN, cnj );

 od;

 od;

 # Verify the stabilizer chain.
 =DoNormalClosurePermGroup java.lang.StringIndexOutOfBoundsException: Index 48 out of bounds for length 48
 if not IsBound( chain.orbits #############################################################################
 if IsBound( chain.aux ) then
#M DerivedSubgroup( <G> ) . . . . . . . ##
 fi;
 result := chain.identity;
 elif random = 1000 then
 missing := chain.missing;
 correct := chain.correct;
 rchain := SCRRestoredRecord( chain );
 result
 if not IsPerm(result) , # list of commutators
 repeat
 result := java.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0
 until result < false;
 fi;
 chain := rchain;

 if $d$ is a prime, <G> is simple
 IsBound(.nownBase random
 return ;
 else
 =[QuoIntd 25920 java.lang.StringIndexOutOfBoundsException: Index 44 out of bounds for length 44
 fijava.lang.StringIndexOutOfBoundsException: Index 11 out of bounds for length 11
 g : ( , g := SCRRandomSubproduct( GeneratorsOfGroup( G ), One(
 param[2] := 2;
 param[ or (d = d 21 Size(G )#

java.lang.StringIndexOutOfBoundsException: Range [18, 14) out of bounds for length 63
 ord SizeAdd list, );
 chain.correct, chain.java.lang.StringIndexOutOfBoundsException: Index 54 out of bounds for length 19
 if java.lang.StringIndexOutOfBoundsException: Range [22, 21) out of bounds for length 36
java.lang.StringIndexOutOfBoundsException: Range [41, 39) out of bounds for length 56

 chain := java.lang.StringIndexOutOfBoundsException: Index 36 out of bounds for length 36
fi
 SetStabChainMutable
 if # if we donStabChainOptionsD).random:=StabChainOptions(G).random;
 N := java.lang.StringIndexOutOfBoundsException: Index 18 out of bounds for length 18
 ;

 # give N the proper randomness
tions()andom=andom;

 # return the normal closure
 UseSubsetRelation( N, U );
 return N;
end );

InstallMethod( NormalClosureOp fi;
IsIdenticalObj, I # do the deterministic method; it will aresult
function(G,U)
 # test applicability
 ifnotIsSubsetGLength (G )) ]
 TryNextMethod();
 fi;
 return60360],,java.lang.StringIndexOutOfBoundsException: Index 56 out of bounds for length 56
end);

#############################################################################
##
#M ConjugateGroup( <G>, <g> ) . . . . . . . . . . . . of permutation group
##
java.lang.StringIndexOutOfBoundsException: Range [76, 13) out of bounds for length 76
function( G, g )
local , S

 H := java.lang.StringIndexOutOfBoundsException: Index 44 out of bounds for length 12
 if HasStabChainMutable) then
 60 :()
 ConjugateStabChain( StabChainMutable( G ), S, g, g );
 SetStabChainMutable( S );
 elif HasSize(G) then
 SetSize(H,Size(G));
 fi;
 UseIsomorphismRelation( G, H );
 ;
end );

#############################################################################
##
#M CommutatorSubgroup( , <V> ) . . . . . . . . . . . . . for perm groups
##
InstallMethod( CommutatorSubgroupfi
 I,
 D :=: or ((t * (4*sm 0 and )java.lang.StringIndexOutOfBoundsException: Index 70 out of bounds for length 70
 local C, # the commutator subgroup
 , closureof ,V
 doneCUV, # boolean; true if CUV is computed
,#
 v, # random subproduct of V.generators
 comm, # commutator of u,v##
 list, # list of commutators
 i; # loop variable

 # [ , <V> ] = normal closure of < [ , <v> ] >.
 C := TrivialSubgroup( U );
 doneCUV := false;

 # if there are lot of generators, use random subproducts
 if Length( # for large degree## J. Symbolic Computation, 12:517--526, 1991.
 Length( GeneratorsOfGroup( V # idea but builds a pcgs on the way).
 repeat
 list :=BindGlobal", local , java.lang.StringIndexOutOfBoundsException: Range [36, 35) out of bounds for length 49
 for i in [1..10] do
 u n ,
 v := SCRRandomSubproduct( GeneratorsOfGroup( V ), One( V ) );
 comm := Comm( u, v );
 not comm Cthen
 ( list,comm ) ;
 ;
 od;
(list)java.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0
 C := ClosureGroup( C, list,
 java.lang.StringIndexOutOfBoundsException: Index 67 out of bounds for length 67
 if not doneCUV# first find a transitive representation for <G>
 =ClosureGroup(, ;
 doneCUVtrue;
 fi;
 # force closure test
 StabChainOptions(C).random:= gens:G;
 : DoNormalClosurePermGroup(CUV, #find commutator
 fi;
 until IsEmptyfijava.lang.StringIndexOutOfBoundsException: Index 11 out of bounds for length 11
 fi;

 # do the deterministic method; it will also check correctness
st = [java.lang.StringIndexOutOfBoundsException: Index 16 out of bounds for length 15
 uinGeneratorsOfGroup U
 for v in GeneratorsOfGroup( homA(fi
 comm := Comm( u, v ); ;
java.lang.StringIndexOutOfBoundsException: Index 11 out of bounds for length 11
list comm ;
 fi;
 od;
 odw
 if not IsEmpty( list ) then
 C:ClosureGroup( C, list, rec( random := 0,
 =java.lang.StringIndexOutOfBoundsException: Index 9 out of bounds for length 9
 if not doneCUV10newjava.lang.StringIndexOutOfBoundsException: Range [29, 28) out of bounds for length 31
CUV=)
 doneCUV := true;
 fi;:,(cannotwhetherjava.lang.StringIndexOutOfBoundsException: Index 10 out of bounds for length 10
 # force closure test
 java.lang.StringIndexOutOfBoundsException: Index 7 out of bounds for length 7
 : returnjava.lang.StringIndexOutOfBoundsException: Index 18 out of bounds for length 18
 fireturn truereturn true;

 return C;
end );

#############################################################################
##
#M DerivedSubgroup( <G> ) . . . . . . . . . . . . . . of permutation group
##
InstallMethod( DerivedSubgroupjava.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0
 function( G )
local ,IsPcgsp then
 g, h, # random subproducts of generators,.permpcgsNormalSteps
 comm, # their commutator java.lang.StringIndexOutOfBoundsException: Index 59 out of bounds for length 59
 # list of commutators
 ;#java.lang.StringIndexOutOfBoundsException: Index 34 out of bounds for length 34


 D:=TrivialSubgroup( G ;

 # if there are >4 generators, use random subproducts#java.lang.StringIndexOutOfBoundsException: Index 72 out of bounds for length 72
 if Length( GeneratorsOfGroup( G ) ) > 4 then
 repeat
 list := [];
 for i in [1..10] do
 g := SCRRandomSubproduct( GeneratorsOfGroup( G ), One
 : SCRRandomSubproduct(GeneratorsOfGroup ,One(G))java.lang.StringIndexOutOfBoundsException: Index 77 out of bounds for length 77
 comm = Comm(g, hjava.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0
 ifor(#

 ;

 if Length(list) > 0 then
ClosureGroup, java.lang.StringIndexOutOfBoundsException: Index 49 out of bounds for length 0
 IsPrimePowerInt( d)then
 # force closure test
java.lang.StringIndexOutOfBoundsException: Range [42, 41) out of bounds for length 57
 (randomjava.lang.StringIndexOutOfBoundsException: Range [61, 60) out of bounds for length 71
 else

 fi;
 D := DoNormalClosurePermGroup( G##
 fi;
 =[;
 fi;

 # do the deterministic method; it will also check random result
 list := [];
 for in [2. Length( GeneratorsOfGroupG) ]java.lang.StringIndexOutOfBoundsException: Range [0, 58) out of bounds for length 22
java.lang.StringIndexOutOfBoundsException: Range [38, 33) out of bounds for length 45
 comm := Comm( IsPcgs(pcgs ()java.lang.StringIndexOutOfBoundsException: Index 36 out of bounds for length 27
 GeneratorsOfGroup( G pcgs;
 if not comm in D then
 Add( list#T form the description why TryPcgs... should also give a drived series. Thus
 fi;
 od;
 od;
 n
 D := ClosureGroup(D,list,rec( random := 0,# ##
 temp := true ) );
 # give D a proper randomness
 ifIsBound((G# if (not DefaultStabChainOptions.tryPcgs
 StabChainOptions(D).random:=StabChainOptions(G).random sIsSolvableGroup()andIsSolvableGroup() then
 else
ptions)random# f IsPcgs(pcgs )
 fi;# fijava.lang.StringIndexOutOfBoundsException: Index 9 out of bounds for length 9

 # if not HasDerivedSubgroup( G ) then
 fi;

 if HasIsNaturalSymmetricGroup(G) and ;

 return # return series;
end );

#############################################################################
##
#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.
##
InstallMethod( IsSimpleGroup," HasIsNilpotentGroup)and ()then
;
 fi
 hom, # transitive constituent or blocks homomorphism
 d, # degree of <G>
 n, m, # $d = n^m$
 simple, # list of orders of simple groups;
 # list of orders of transitive perfect groups
 s, if not HasDe( ) then

 # if <G> is the trivial group, it is not simple
 if IsTrivial( :f;
 return java.lang.StringIndexOutOfBoundsException: Range [18, 17) out of bounds for length 18
java.lang.StringIndexOutOfBoundsException: Index 7 out of bounds for length 7

 # first find a transitive representation for <G>
 D: ( ,SmallestMovedPoint(G )
 if not IsEqualSet( MovedPoints( Gnew:=Comm(ens[i],nd and IsFinite)
 hom := ActionHomomorphism( G, D,"surjective" );
 Size j+;
 return false;
 fi;
 G pcgsjava.lang.StringIndexOutOfBoundsException: Index 11 out of bounds for length 11
 fi;

 # next find a primitive representation for <G>
 : (G,MovedPoints G ))java.lang.StringIndexOutOfBoundsException: Index 39 out of bounds for length 39
 while Length( D ) <> java.lang.StringIndexOutOfBoundsException: Index 25 out of bounds for length 14
 hom := ActionHomomorphism( G, D, OnSets,java.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0
 ( )<> Size(Image( hom# [ IsList ], 0,
 return false;
 java.lang.StringIndexOutOfBoundsException: Index 11 out of bounds for length 11
 a:Random(l^b=(l,#( list[ 1])
 D := Blocks(newC## fi;
 od;

 if not IsPcgs( pcgs ) then return fail;
 D := MovedPoints( G );
 d := Length(
 n := java.lang.StringIndexOutOfBoundsException: Range [0, 24) out of bounds for length 17
 m := LogInt( d, n );
 if 10^6 < ifnew=InstallMethod(,"or,true,[IsPermGroup] 0java.lang.StringIndexOutOfBoundsException: Index 82 out of bounds for length 82
 Error("cannot decideifF>java.lang.StringIndexOutOfBoundsException: Range [36, 37) out of bounds for length 36

issimple
 G
 return truejava.lang.StringIndexOutOfBoundsException: Index 19 out of bounds for length 11

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

 # if $G = S_d$, it is not simple (except $S_2$) java.lang.StringIndexOutOfBoundsException: Range [43, 30) out of bounds for length 58
elifjava.lang.StringIndexOutOfBoundsException: Range [34, 33) out of bounds for length 41
 return 2 = ;

 # if $G$ is not perfect, it is not simple (unless $G = C_p$, see above)SetPcgsElementaryAbelianSeries,cgs;
 ( ;
 return false;

 # if $\|G\| = d^2$, it is not simple (Kantor's Lemma 4)
 elif Size fi S=GroupStabChain , )) ;
 return false;

 # if $d$ is a prime, <G> is simple
 elif Group S, true );
 return true;

 # if $G = U(4,2)$, it is simple (operation on 27 points)
Size( G) = ", true, [IsPermGroup ], 0,
 return true;

 # if $G = PSL(n,q)$, it is simple (operations on prime power points)
 (##
 or (d )
 or (java.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0
 ) # PSL(5,3)
 or (d = 128 and S # -Sylow subgroup of <G>, result(G)
 d= SizeG) 8191^8191)2) # PSL(2,8191)
 D, # domain of operation of <G>
 or (d = 524288 and Size(G) (^-524287) blocks of the operation of <G> on <D>
 and IsTransitive( Stabilizer( SetIsPcgsCentralSeries(pcgs,true)
 then
 return true;

 # if $d$ is a prime power, <G> is not simple (except the cases above)
 elif q := 1; java.lang.StringIndexOutOfBoundsException: Index 5 out of bounds for length 5
 return false;

 # if we don't have at least an $A_5$ acting on the top, <G> is simple
 elif m < 5 then
 return true;

 # otherwise we must check for some special cases
java.lang.StringIndexOutOfBoundsException: Index 20 out of bounds for length 8

simplelocalpcgs;
 simple := [ ,,,,,
 [60,360],,,, # 5: A(5), A(6)
 [60,360,1814400],, # 10: A(5), A(6), A(10)
 [660,7920,95040,239500800],, # 12: PSL(2,11), M(11), M(12), A(12)java.lang.StringIndexOutOfBoundsException: Index 30 out of bounds for length 7
 [1092,43589145600], # 14: PSL(2,13), A(14)
fijava.lang.StringIndexOutOfBoundsException: Index 7 out of bounds for length 7
 )

 # orders of transitive perfect subgroups of $S_m$ repeatg: Random ;until ( ) p= 0ved .Thus
 transperf := [ ,,,,
 [60], # 5: A(5)
 [60,360], # function( G )
 [168,2520], # 7: PSL(3,2), A(7)
 [168,8168,20160] # 8: PSL(3,2), AGL(3,2), A(8)
 ]java.lang.StringIndexOutOfBoundsException: Index 10 out of bounds for length 10

 # test the special cases (Kantor's Lemma 3)
#
 for # TryNextMethod();
 if # SetIndicesEANormalSteps(( ;
 # if Length( series ) > 1 then SetDerivedSubgroup( G, series[ 2 ] );
 or (
 then
 return false;
 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.
(java.lang.StringIndexOutOfBoundsException: Range [39, 38) out of bounds for length 51
local TryNextMethod
 # a few moved points
 som:=MovedPointspcgs: f :ActionHomomorphism B,OnSetssurjective")
( 0java.lang.StringIndexOutOfBoundsException: Range [26, 25) out of bounds for length 64
 if not HasDerivedSubgroup java.lang.StringIndexOutOfBoundsException: Index 33 out of bounds for length 21
 if Length(som)>100 then
java.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0
 fi;

 elvth:=One(G);
 fct=unction()
 elvth:=elvth*Product([1..Random(1,5)],x->Random(GeneratorsOfGroup(G))^java.lang.StringIndexOutOfBoundsException: Index 80 out of bounds for length 57
 return elvth;
 ;

=(G;
 # find one nontrivial commutator
 new:=fail;
 i:=1; g2 :RandomC) ( )p=0;
 while i<=Length(gens) and new=fail do
 j:i+
 while j<=Length( C2 := Centralizer)
 ifForAny(-(^[]^gensj]>x[j]^ens]
 new:=Comm(gens[i], C := C2; g := g2;
 od
 j:java.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0
 od;
i=+;
 od;
 if new=fail then return I,,PermSylow ";

 # now take commutators with conjugates to get down
 i:=1; = (f), java.lang.StringIndexOutOfBoundsException: Index 49 out of bounds for length 49
 repeat
 gens:=;
 j:=1;
 new:=fail;
 l:=[gens];
 #############################################################################
 a=Random()^fct();
 b:=First(l,b->ForAny(som,x->java.lang.StringIndexOutOfBoundsException: Index 2 out of bounds for length 2
 if b<>fail then
new=(a,b)java.lang.StringIndexOutOfBoundsException: Index 23 out of bounds for length 23
 else
 Addl,java.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0
 fi;
 j:=j+1;
 odIsPrimitive,)
return (java.lang.StringIndexOutOfBoundsException: Index 24 out of bounds for length 24
 return fail;
 java.lang.StringIndexOutOfBoundsException: Index 6 out of bounds for length 0
 :i+;
 until i>bound;
 return true;
end);

InstallMethod( IsSolvableGroup,"for permgrp", true, [ IsPermGroup ], 0,
function(G)
local pcgs;
 if QuickUnsolvabilityTestPerm(G)# and not IsSolvableGroup( list[ TryNextMethod)

 pcgs:=TryPcgsPermGroup#java.lang.StringIndexOutOfBoundsException: Index 35 out of bounds for length 35
 if IsPcgs(pcgs) then
 SetIndicesEANormalSteps(pcgs,pcgs!.permpcgsNormalSteps);
 SetIsPcgsElementaryAbelianSeries(pcgs,rue);
 if not HasPcgs(G) then
 java.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0
 fi;
 if not HasPcgsElementaryAbelianSeries(G) ##
 SetPcgsElementaryAbelianSeries(G,pcgs);
 fi;
 return true;
 # m := m * c[ 1 ] ^ ( c[ 2 ] / d );
 return false;
 ;
;

#############################################################################
##
#M IsNilpotentGroup( <G> ) . . . . . . . . . . . . . . . . . nilpotency test
##
InstallMethod( IsNilpotentGroup,"for java.lang.StringIndexOutOfBoundsException: Index 12 out of bounds for length 12
 G -> IsPcgs : java.lang.StringIndexOutOfBoundsException: Index 21 out of bounds for length 20

#############################################################################
##
#M PcgsCentralSeries( <G> )
##
InstallOtherMethod( PcgsCentralSeries,"for permgrp
function(G)
local pcgs;
G,true
 if IsPcgs(pcgs) #M SylowSubgroup( <G>, i : java.lang.StringIndexOutOfBoundsException: Range [19, 18) out of bounds for length 23
 (pcgs,pcgs!permpcgsNormalSteps);
 SetIsPcgsCentralSeries(pcgs,true);
 if not[IsPermGroup,
 etPcgsG,java.lang.StringIndexOutOfBoundsException: Index 22 out of bounds for length 22
 fi;
 fi;
 return pcgs # enforce solvable methods if we detected anew that the group is
end);

#############################################################################
##
#M PcgsPCentralSeriesPGroup( <G> )
##
java.lang.StringIndexOutOfBoundsException: Range [33, 18) out of bounds for length 85
)
local Last;
 java.lang.StringIndexOutOfBoundsException: Index 7 out of bounds for length 7
 if
 java.lang.StringIndexOutOfBoundsException: Index 8 out of bounds for length 6
 SetIsPcgsPCentralSeriesPGroup(pcgs,true);
 fi
 if IsPcgs#M FrattiniSubgroup( <G> q# largest of p>dividing the size##
SetPcgs(,)java.lang.StringIndexOutOfBoundsException: Index 20 out of bounds for length 20
 fijava.lang.StringIndexOutOfBoundsException: Index 5 out of bounds for length 5
;
endB

#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
##
,,Ggens

 local pcgs, seriesOrbitsDomainStabilizerpunkt,

 return [G g = g ( g) /p# orbit:=Difference(StabChainMutable(G).orbit,[punkt]);table().orbit,punkt]);
 fi;

 # stab:=[ one ];
 or HasIsNilpotentGroup
 and
 TryNextMethod# processStabGen:=function()
 fi;

 pcgs := StabGenAvailable:=false;
 if not IsPcgs( pcgs ) then
 TryNextMethod(# end;
 fi;
 SetIndicesCentralNormalSteps(,pcgs.permpcgsNormalSteps);
 series : = java.lang.StringIndexOutOfBoundsException: Index 15 out of bounds for length 15
 if not HasDerivedSubgroup( G ) then
 if Length( series ) > 1 punkt]: onejava.lang.StringIndexOutOfBoundsException: Index 26 out of bounds for length 26
 elseforgen do
 fi;
 return series;
end =SylowSubgroupPermGroup( Range( ), ]: gen;

 (, T ;
[ IsPermGroup and ;
 # we want this to come *before* the method for Pcgs computable groups;
 RankFiltercurrPt=
 -currGen0
function(Gjava.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0
local pcgs;
 pcgs(;
 if B:= Blocks(G ;
 TryNextMethod();
;
 return EANormalSeriesByPcgs Length)java.lang.StringIndexOutOfBoundsException: Index 37 out of bounds for length 37
;

#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;
function( G, p java.lang.StringIndexOutOfBoundsException: Index 9 out of bounds for length 9
 local pcgs;

 if Length(Factors(Size(G))# deg > 3125
 TryNextMethod;
 fi;
 pcgs:=PcgsPCentralSeriesPGroup(G);
 not(
 TryNextMethod(# if d mod 5 = 0 then
 fi;
 return PCentralNormalSeriesByPcgsPGroup
end);

#############################################################################
##
#M SylowSubgroup( <G>, ) . . . . . . . . . . . . . . Sylow $p$-subgroup
##
InstallMethod( SylowSubgroupOp,"permutation
 [ #############################################################################
function( G, p )
local S;
 if not HasIsSolvableGroup(G) InstallGlobalFunction( SignPermGroup(g
#
 # solvable.
 SylowSubgroupOp)
 :[java.lang.StringIndexOutOfBoundsException: Index 10 out of bounds for length 10
 S := SylowSubgroupPermGroupcoll ] 2] j1
S =(,StabChainMutable ;
 SetIsNilpotentGroup( S, true );
 if Size(S) > 1 then
 m-1:m[-1]+
 SetPrimePGroup( S, p );

 fi m-1]:=1
 return S;
end );

InstallGlobalFunction( SylowSubgroupPermGroup, function( G, p )
 local S, # -Sylow subgroup of <G>, result

 D, # domain of operation of <G>end )
 O,#java.lang.StringIndexOutOfBoundsException: Index 57 out of bounds for length 57
 B, # speed up the closure process##
 f homomorphism of G> java.lang.StringIndexOutOfBoundsException: Range [0, 55) out of bounds for length 2
 T, # -Sylow subgroup in the image of <f>
 , g2 # one element of <G>
 C # centralizer of <g> in <G>

 # get the size of the -Sylow subgroup m:=([1..engthl1]i>);
 q := 1; while Size( G ) mod (q * p) = 0 do q := q * p; od;

 # handle trivial subgroup
 if q #############################################################################
 return TrivialSubgroup( G );
 fi##
 # go down in stabilizers as long as possible
 S := StabChainMutable( G );
 whileLength(Sorbit ) mod p <> 0 do
 S := S.stabilizer;
 od;
 G := t )

 # handle full group
 if q = Size( G ) then
 return G;
fi

 # handle -Sylow subgroups of size 
 if =pthen
 repeat g := Random( G ); l:=GeneratorsOfGroup(G);
 g := g ^ (Order( g ) / p);
 return SubgroupNC( G, [ g ] );
 fi;

 # if the group is not transitive work with the transitive constituents
 D := MovedPoints( G );
 java.lang.StringIndexOutOfBoundsException: Range [23, 6) out of bounds for length 37
 Info(InfoGroup (,l[i]^p)java.lang.StringIndexOutOfBoundsException: Index 23 out of bounds for length 23
 S := G;
 D: ShallowCopy )java.lang.StringIndexOutOfBoundsException: Index 30 out of bounds for length 30
 while q < Size( S ) do
 O := Orbit( S, D[1] );
 f := #M ApproximateSuborbitsStabilizerPermGroup(<G>,<pnt>) . . . approximation of
 T := SylowSubgroupPermGroup( Range( f ), p );
 S := PreImagesSet( f, T );
 SubtractSet( D, O );
 od;
 return S;
 fi;

 # if the group is not primitive work in the image first
 B := Blocks( G, D );
 ifLength( B ) <> 1 then
 Info(InfoGroup,1,"PermSylow: blocks");
 f = ActionHomomorphism( G,B, OnSets,surjective" );
 T := SylowSubgroupPermGroup( Range( f ), p );
 if Size( T ) < Size( Range( f ) ) then
 T := PreImagesSet( f, T );
 S:=SylowSubgroupPermGroup( T , p) ;
 return S;
 fi;
 fi;

 # find a element whose centralizer contains a full -Sylow subgroup# stabilizer, a trivial orbit algorithm seems best.
 Stabilizer(,unkt),
 repeat g :Difference(MovedPoints(),punkt]))java.lang.StringIndexOutOfBoundsException: Index 64 out of bounds for length 64
 g := g ^# stab:=Stabilizer(G,punkt));
 C := Centralizer( G, g );
 while GcdInt( q, Size( C ) ) < q do
 repeat g2 := java.lang.StringIndexOutOfBoundsException: Range [0, 27) out of bounds for length 13
 g2 :# if stgp>Length(stab) then
# return stab[stgp-1];
 if GcdInt( q, Size( C ) ) < GcdInt( q, Size( C2 ) ) then
 C :
 fi # compute orbit and transversal
 od;

 # the centralizer acts on the cycles of the element
 for pnt indo
 Info(InfoGroup,1,"PermSylow img:pnt^genjava.lang.StringIndexOutOfBoundsException: Index 21 out of bounds for length 21
 f := ActionHomomorphism( C, java.lang.StringIndexOutOfBoundsException: Range [8, 33) out of bounds for length 11
 T := SylowSubgroupPermGroup( Range( f ), p );
 S := PreImagesSet( f, od;
 return S;

end );

#############################################################################
##
#M Socle( <G> ) . . . . . . . . . . . socle of primitive permutation group
##
InstallMethod( Socle: One g );
 for i in Conjug i (g do
 , ,coll L ,ordjava.lang.StringIndexOutOfBoundsException: Index 50 out of bounds for length 50
 p, i;

 Omega := MovedPoints( G

 if not IsPrimitive( G,
 TryNextMethod();
 turn e is a^e=b or false otherwise

 # Affine groups first.
);
Ljava.lang.StringIndexOutOfBoundsException: Index 20 out of bounds for length 20
 return L[1];
 fi

 deg := Length( Omega
 if deg >= 6103515625 then
;
 # the normal closure actually seems to be faster than derived series, so
 # this is not really a shortcut
#
 # shortcut := true;
 # if deg >= 3125 then
# coll := Collected( Factors( deg ) );
 = (coll >c 2]))
 # if d mod 5 = 0 then ,abgjava.lang.StringIndexOutOfBoundsException: Index 28 out of bounds for length 28
 m : 1java.lang.StringIndexOutOfBoundsException: Index 24 out of bounds for length 24
 # for c in coll do
 # m := m * c[ 1 ] ^ ( c[ 2 ] / d );
 :;
 # if m >= 5 then
 # shortcut := false;
 # fi;
 # fi;
 # fi;
 # if shortcut then
 # ds := DerivedSeriesOfGroup( G );
 # return Last(ds);
 # fi;
 java.lang.StringIndexOutOfBoundsException: Index 7 out of bounds for length 7

 gens (G);
 deg <78125
 # mdescendingorder;trivial
 else
 i := Length( coll );
 while coll[ i ][ 2java.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0
 else
 odgen:=b]java.lang.StringIndexOutOfBoundsException: Index 22 out of bounds for length 22
 p :[ ] ]
 fi;

 repeat
 z := Random( G );
 : Order( )java.lang.StringIndexOutOfBoundsException: Index 30 out of bounds for length 30
 ord p=0java.lang.StringIndexOutOfBoundsException: Index 28 out of bounds for length 28
 =z^ (/java.lang.StringIndexOutOfBoundsException: Range [0, 26) out of bounds for length 5
 until Index( Lengthgens)=()2

 # immediately add conjugate as this will be needed anyhow and seems togen<orbit[]then (") java.lang.StringIndexOutOfBoundsException: Index 57 out of bounds for length 57
 # speed up the closure process
 L := NormalClosure( G, SubgroupNC( java.lang.StringIndexOutOfBoundsException: Index 1 out of bounds for length 0
 if deg >= 78125 then
 ds := # Schreier generators
 =Last(ds)java.lang.StringIndexOutOfBoundsException: Index 22 out of bounds for length 22
 changed=3 and currPt<=Length( do;
 if IsSemiRegular( L, Omega ) then
 : ClosureSubgroup ,(G,L);
 fi;
 L;
end );

#############################################################################
##
#M FrattiniSubgroup( <G> ) . . . . . . . . . . . . for permutation p-groups
##
(java.lang.StringIndexOutOfBoundsException: Range [33, 31) out of bounds for length 72
 java.lang.StringIndexOutOfBoundsException: Index 17 out of bounds for length 17
 localodjava.lang.StringIndexOutOfBoundsException: Index 5 out of bounds for length 5

 if not IsPGroupjava.lang.StringIndexOutOfBoundsException: Range [11, 10) out of bounds for length 27

 elifmintry=
 return ()2 then
 fi;
 p := PrimePGroup( G );
 l := GeneratorsOfGroup( G#############################################################################
 k := [ l[1]^p ];
..Length(l)]
 for j in [ 1 .. i-1 ] ifmintry=(ens return gens;fi;
 Add(k, Comm(l[i] , ,java.lang.StringIndexOutOfBoundsException: Index 40 out of bounds for length 40
 od;
 Add(k, l[i]^p);while<5iLengthjava.lang.StringIndexOutOfBoundsException: Index 34 out of bounds for length 21
odjava.lang.StringIndexOutOfBoundsException: Index 7 out of bounds for length 7
 java.lang.StringIndexOutOfBoundsException: Range [44, 41) out of bounds for length 50
end );

 j:=j+1;
##
#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( ApproximateSuborbitsStabilizerPermGroup,
 function,unkt)
 local one, orbit, trans, gen, pnt,
 processStabGen, currPt,currGenF(n1,,i

 one:= One( G
 if HasStabChainMutableAdd
# if we already have a stab chain and bother computing the proper
# stabilizer, a trivial orbit algorithm seems best.
 fi
 OrbitsDomain(Stabilizer(G,java.lang.StringIndexOutOfBoundsException: Index 45 out of bounds for length 14
 (MovedPoints()[unkt)
# 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;

 java.lang.StringIndexOutOfBoundsException: Range [6, 7) out of bounds for length 6
 # now get the smallest generating system of the stabilizer
 Ggens:=GeneratorsOfGroup(G);
 orbit := [ punkt ];
 trans := [];
 trans[punkt ] :one;
 for pnt in orbit do
forgenin Ggens do
 img:=pnt^gen;
 if not SubtractSet( l=[;
 java.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0
 trans[img ] :;
 ;
 ;
 od;

 fi;:1]java.lang.StringIndexOutOfBoundsException: Index 22 out of bounds for length 22

 currPt:=1;
 currGen: ( S.[ ] )java.lang.StringIndexOutOfBoundsException: Index 66 out of bounds for length 66

=
 localpunkt,mg,b stabilizer
if(,)1
 # next one
 Add,mjava.lang.StringIndexOutOfBoundsException: Index 13 out of bounds for length 13
 if currGen ;
 :1java.lang.StringIndexOutOfBoundsException: Index 21 out of bounds for length 21
 currPt:=currPt+1;
 if;
 return ();
 fi;
 fi;
 a:=currPt;
 b:=currGen;
 else
 a:=Random(1, Length(orbit));
 b=Random(1,LengthGgens)java.lang.StringIndexOutOfBoundsException: Index 36 out of bounds for length 36
 PositionlCRepresentative))1]=;
 # map a with b
 img:=orbit[a]^Ggens[b java.lang.StringIndexOutOfBoundsException: Range [22, 21) out of bounds for length 30
img
 return ();
 else
 punkt:=orbit[a];
 gen:=Ggensjava.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0
 while punkt<,,,cp,g
 gen: return the smallest generating system
 punkt:punkt/trans[punkt];
 od;
 if orbit1]java.lang.StringIndexOutOfBoundsException: Index 1 out of bounds for length 0
 punkt:=mg;
 while punkt<>orbit[1] do
 gen=/ranspunkt;

 odjava.lang.StringIndexOutOfBoundsException: Index 11 out of bounds for length 11
 if orbit[1]^gen<while m<ob do
 fi;
 return java.lang.StringIndexOutOfBoundsException: Range [32, 31) out of bounds for length 53
 end;

 # as the first approximation take a subgroup generated by four nontrivial
 # Schreier generators
 changed=0;
java.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0
 changed<=3 and currPt<=Length(orbit) do
 g:=Gcdm,Length(c));
 while or(e modgjava.lang.StringIndexOutOfBoundsException: Range [26, 19) out of bounds for length 19
 gen:processStabGen();
 od;
 Addgens,gen);
 changed:=changed+1;
 od#M SmallGeneratingSet(<G>)
if(gens)0
 : : GeneratorsOfGroup ;
 else
 orbit=OrbitsPermsgensorbit;
 fi;

 # todo: further fuse
 return orbit;
end );

#############################################################################
##
#M AllBlocks . . . Representatives of all block systems
##
InstallMethod(AllBlocks, (strPluralize(( gens )");
function(g)
localjava.lang.StringIndexOutOfBoundsException: Index 6 out of bounds for length 6

 DoBlocks(b
 local bl,bld,i,t,n;
 find representatives of all potential suborbits
 bld:=List(Filtered(subo,i->not ForAny(b,j->j in i)),i->i[1]);
 bl:=[];
 a=(ens] gens[ fi;
 for i in bld do
 t:=Union(b,[i]);
 nj:= +1java.lang.StringIndexOutOfBoundsException: Index 19 out of bounds for length 19
 if Length(n)>1 and
 not n[1] in pool then
 ##
 #M AsList( <G> ) elements of perm group
 bl:=InstallMethod(AsList,"permgp: ElementsStabChain,[IsPermGroup ]
 fi;
 od;
 fi;
 ;
 end;

 one:= One( g );
< java.lang.StringIndexOutOfBoundsException: Index 57 out of bounds for length 57
if(gens)=then return [;fi;
 subo:=ApproximateSuborbitsStabilizerPermGroup(g,
 Minimum(List( java.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0
 IsSolvableGroupG)
otg,) java.lang.StringIndexOutOfBoundsException: Range [33, 34) out of bounds for length 33
 Sort(list
 ;
 pool:=[endjava.lang.StringIndexOutOfBoundsException: Index 6 out of bounds for length 6
 return DoBlocks(subo[############################################################################
end);

#############################################################################
##
#F SignPermGroup
##
InstallGlobalFunction LengthG)Length=);
 if ForAll( IsPrimitived)then
 Error(<Gprimitive"
 else
 return -1;
 fi;
end );

java.lang.StringIndexOutOfBoundsException: Range [13, 10) out of bounds for length 51
java.lang.StringIndexOutOfBoundsException: Range [6, 5) out of bounds for length 14
 return Size(U)=Size(G);
 for i in Partitions(n)return 5;
 m:=[];
 for java.lang.StringIndexOutOfBoundsException: Range [13, 12) out of bounds for length 74
 if j>1 then
j-1] java.lang.StringIndexOutOfBoundsException: Range [0, 31) out of bounds for length 26
 m[j-1]:=m[j-1]+1;
 else
 m[j-1]:=1;
 fi;
 fi;

 Addif()-]java.lang.StringIndexOutOfBoundsException: Index 24 out of bounds for length 24
 od;
 return l;
end

#############################################################################
##
#F CycleStructuresGroup(G) list of all cyclestructures occ. in a perm group
##
InstallGlobalFunction( CycleStructuresGroup, java.lang.StringIndexOutOfBoundsException: Index 52 out of bounds for length 22
one
 l:=CreateAllCycleStructures(Length(java.lang.StringIndexOutOfBoundsException: Index 46 out of bounds for length 42
 m ;;
 one )
 for in # power of the
java.lang.StringIndexOutOfBoundsException: Range [22, 21) out of bounds for length 36
 m[Position(l,CycleStructurePerm(Representative(i)))-1]:=1;
 fi;
 od;java.lang.StringIndexOutOfBoundsException: Range [15, 14) out of bounds for length 20
 return m;
end );

# return e is a^e=b or false otherwise
InstallGlobalFunction("LogPerm",
function(a,b)
java.lang.StringIndexOutOfBoundsException: Range [16, 5) out of bounds for length 33
 oa:= :=(U);
 ob:=Order(b);
 if oa mod ob<>0 then Info(InfoGroup i:=i1java.lang.StringIndexOutOfBoundsException: Index 13 out of bounds for length 13
 gens;
 aa:=a^java.lang.StringIndexOutOfBoundsException: Range [0, 9) out of bounds for length 5

 m:=1; # order modulo which power is OK so far
 e:=1;
 while m<ob do
 tst:=aa^e/b;
 s:=SmallestMovedPoint(DeclareGlobalFunction "eneratorsSmallestStab );
 if s=infinity then # found it
 returnInstallGlobalFunction(GeneratorsSmallestStab, function ( S )
 fi;
 c:=Cycles);
 g:=Gcd(m,Length(c));
 p:=Position(c,s^b);
 if p=fail or (e mod g< orb
 e:=ChineseRem([m,Length(c)][e,-]java.lang.StringIndexOutOfBoundsException: Index 41 out of bounds for length 41
 m:=Lcm(m,Length(c));
 od;
 if aa^e<>b then return false;fi;
 return q*e mod oa;
end);

#############################################################################
##
#M SmallGeneratingSet(<G>)
##
InstallMethod(SmallGeneratingSet,"random and generators subset, randsims",true,
 [IsPermGroup],0,
function (G)
local i, j, U, gens,a,mintry,orb,orp,isok;

 gens := ShallowCopy(Setjava.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0

 # remove obvious redundancies...
 # sort elements by descending order; trivial permutations
 # at the end
SortBy(gens, g-> Order(g);

 i := 1;theother points in the orbit
 while i < Length(gens) do
 j := i + 1;
 while j <= Length(gens do
 a:=LogPerm(gens[i], gens[j]);
 if a = false then
 j := j + 1;
 else
 Remove(gens, j);
fi;
 od;
 i := i + 1;
 od;

 if Length(gens)<=Length(java.lang.StringIndexOutOfBoundsException: Index 28 out of bounds for length 26
 return pntOnTuples orbitgen)
 fi;

 # try pc methods. The solvability test should not exceed cost, nor

 if #Length(MovedPoints(G))<50000 and
 #((HasIsSolvableGroup(G) and IsSolvableGroup(G)) or IsAbelian(G)) =java.lang.StringIndexOutOfBoundsException: Range [30, 29) out of bounds for length 30
 IsSolvableGroup(G)
andLengthgens> java.lang.StringIndexOutOfBoundsException: Index 29 out of bounds for length 29
 return MinimalGeneratingSet
 fi;

 # store orbit data
 orb:=Set(Orbits(G,MovedPoints(G)), :[.1]
 # longer primitive orbits
 orp:=Filtered([1..Lengthjava.lang.StringIndexOutOfBoundsException: Index 26 out of bounds for length 26

 isok:=function(U)
 if Length(MovedPoints(G))>)java.lang.StringIndexOutOfBoundsException: Index 6 out of bounds for length 6
 Length(orb)>Length(Orbits(U,MovedPointsIsPermGroup,
 (xUx))then
 return false;
 fi;

 (U
 return ()S(G)java.lang.StringIndexOutOfBoundsException: Index 27 out of bounds for length 27
 end;

 :2;
 if java.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0

 # structure of G/G', as the latter is more expensive.
 mintry:=Maximum(java.lang.StringIndexOutOfBoundsException: Range [0, 24) out of bounds for length 2
 mintry:=Maximum(mintry,2);
 if mintry=Length(gens) then return gens;fi;
 :=aximum2m "
 while i<=mintry+1 and i< (,tringjava.lang.StringIndexOutOfBoundsException: Range [29, 28) out of bounds for length 34
 # try to find a small generating system by random search>java.lang.StringIndexOutOfBoundsException: Index 20 out of bounds for length 20
 j:=1;
 whilestr"";
 U:=Subgroup(G,List([1..i], Append(str,ViewString(gens));
 if isok(U) then
 gens:=Set(GeneratorsOfGroup(U));
 str;
 j:=j+1end
 od;
 i:=i+1;
 od;
 fi##

 # try to delete generators
 i := 1;
 while i < (ens)and Lengthgens>mintry do
 # random did not improve much, try removing generators one by one
 U:=Subgroup(G,gens{Difference([1..Length(gens)],[i])});
 ifisok(U then
 gens:=Set(GeneratorsOfGroup(U));
 else
 i:=i+1;
 fi;
 od;
 return gens;
end);

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

InstallGlobalFunction( GeneratorsSmallestStab, function ( java.lang.StringIndexOutOfBoundsException: Range [0, 59) out of bounds for length 13
 localend )
 gen, # one generator in <gens>
 ##
 pnt, # one point in <orb>
 T, # stabilizer in <S>
 o2,img,oo,og; # private orbit algorithm

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

 # now get the smallest generating system of the stabilizer
 gens: GeneratorsSmallestStab( S.stabilizer );

 # get the sorted orbit (the basepoint will be the first point)
orb=Set(Sorbit );
 SubtractSet( java.lang.StringIndexOutOfBoundsException: Range [0, 20) out of bounds for length 15

 # handle the other points in the orbit
 while Length(orb) <> 0 do

 # take the smallest point (coset) and one representative
 pnt := orb[1];
 gen : fi
 while S.orbit[1] ^ gen <> pnt do
java.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0
 od;

 :Stabilizer(,(s)java.lang.StringIndexOutOfBoundsException: Index 45 out of bounds for length 45
 T := gth()1]
 while(Tgenerators)< do
 pnt := Minimum( OnTuples#M SocleTypePrimitiveGroup( <G> )
 while T.orbit[1] ^ gen <> pnt do
 gen := LeftQuotient( T.transversal[ pnt / gen ], gen ) # now test Dixon&Mortimer, p.126, Case 1: R1 (projection of pt. stab
 od;
 T := T.stabilizer;
 od;


 Add( gens, gen );

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

#
 o2:=[S.orbit[1]];
 RemoveSet(orb,S.orbit[1]);
 for oo in o2 do
 forog in do
 img:=oo^og;
 if img in orb then
 Add(o2,img);
 java.lang.StringIndexOutOfBoundsException: Range [8, 9) out of bounds for length 8
 fi;
 od;
 od; b;

 od;

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

InstallMethod(GeneratorsSmallest,"##
 [#M SocleTypePrimitiveGroup( <G> )
function(G)

 return GeneratorsSmallestStab(java.lang.StringIndexOutOfBoundsException: Index 39 out of bounds for length 28
cs(;

InstallMethod(LargestElementGroup,"perm group via minimal stab chain",
 [IsPermGroup],
 # call the recursive function to do the work
 G -> LargestElementStabChain( MinimalStabChain( G ), One( G ) ) );

#############################################################################
##
#M KnowsHowToDecompose( <G>, <gens> )
##
# this function is needed for the transitive and primitive groups libraries
 "perm group and generators: always true",
 IsIdenticalObj,
 [ IsPermGroup, IsList return b(a);
 ReturnTrue);

#############################################################################
##
#M ViewObj( <G> )
##
InstallMethod( ViewString
 [ IsPermGroup and HasGeneratorsOfGroup java.lang.StringIndexOutOfBoundsException: Index 43 out of bounds for length 4
function(G)
 local gens,str;
 gens:= GeneratorsOfGroup( G );
 if 30 < Length(d:List(i>g);
 str<java.lang.StringIndexOutOfBoundsException: Range [23, 22) out of bounds for length 30
 if HasSize(G) then
 Append(str," of size ");
str,Size(G)java.lang.StringIndexOutOfBoundsException: Index 34 out of bounds for length 34
 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
 Append(str,"()");
 else
 (,iewString((gens)java.lang.StringIndexOutOfBoundsException: Index 35 out of bounds for length 35
 fi;
 Append(str,")");
 fi;
 return str;
end);

#############################################################################
##
#M AsList( <G> ) elements of perm group
##
InstallMethod( AsList, "permgp: ElementsStabChain" ;
 G -> ElementsStabChain( StabChainMutable(G)) );

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

InstallMethod( AsSSortedListNonstored, "via stabchain", [IsList)=then
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,ctmjava.lang.StringIndexOutOfBoundsException: Index 23 out of bounds for length 23
 dom:=MovedPoints(G);
 if not IsPrimitive(G,dom) then
 Error("<G> must be primitive");
 fi;
 s:=Socle(G);
 if IsAbelian(s) then
 return "1";
 elif IsNonabelianSimpleGroup(s) then
 return "2";
s then
 return "5";
 else
 # 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:=Stabilizer(s,SmallestMovedPoint(s));
 cs:=CompositionSeries(s);
 t:=cs[Length(cs)-1];
 m:=cs[2];

 # now test Dixon&Mortimer, p.126, Case 1: R1 (projection of pt. stab
 # onto one direct factor) is proper subgroup of T1.:=+;

#java.lang.StringIndexOutOfBoundsException: Index 49 out of bounds for length 49
 while not ForAny(orb[p],j->ForAny(GeneratorsOfGroup(s),k->j^k<>j)) do
 # act transitively
 if p:=p1;
 return "4c"; # Product action of type 2 with transitiveifp>Length(orb then :;ijava.lang.StringIndexOutOfBoundsException: Index 36 out of bounds for length 36
 fi;

 # Now we are in case 2, R1=T1. Group must be diagonal (3) or product ndom=Unionndom,[p)java.lang.StringIndexOutOfBoundsException: Index 29 out of bounds for length 29
 # action of diagonal with transitive (4). For diagonal case the point
 # stabilizer must have the order of t, for product case it must be a
 od
 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(G,t))<Length(cs)-1 then
 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
 return "4b";
 fi;
 fi;
 fi;
end);

#############################################################################
##
#M SocleTypePrimitiveGroup( <G> )
##
InstallMethod(SocleTypePrimitiveGroup,"primitive permgroups",true,
 [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);
 cs:=CompositionSeries(s);
 t:=cs[Length(cs)-1];
 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
 return b(a);
 elif IsList(b) and not IsString(b) then
 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(d,i));
 u:=List(GeneratorsOfGroup(g),i->Product(ran,j->Image(emb[j],i)));
 u:=SubgroupNC(d,u);
 act:=ActionHomomorphism(d,RightTransversal(d,u),OnRight,"surjective");
 return Image(act,d);
end);

InstallGlobalFunction(ReducedPermdegree,function(g)
 local dom, orb, gdeg, p, ndom, s,hom;
 dom:=MovedPoints(g);
 orb:=ShallowCopy(Orbits(g,dom));
 if Length(orb)=1 then return fail;fi;
 gdeg:=20*LogInt(Size(g),2); # good degree
 SortBy(orb, Length);
 p:=PositionProperty(orb,i->Length(i)>=gdeg);
 if p=fail then p:=Length(orb);fi;
 ndom:=orb[p];
 if Length(ndom)*2>Length(dom) then return fail;fi;
 s:=Stabilizer(g,orb[p],OnTuples);
 p:=p+1;
 if p>Length(orb) then p:=1;fi;
 while Size(s)>1 do
 while not ForAny(orb[p],j->ForAny(GeneratorsOfGroup(s),k->j^k<>j)) do
 p:=p+1;
 if p>Length(orb) then p:=1;fi;
 od;
 ndom:=Union(ndom,orb[p]);
 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);

Messung V0.5 in Prozent

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PVS Prover

Isabelle Prover

NIST Cobol Testsuite

Cephes Mathematical Library

Vienna Development Method

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Die Informationen auf dieser Webseite wurden nach bestem Wissen sorgfältig zusammengestellt. Es wird jedoch weder Vollständigkeit, noch Richtigkeit, noch Qualität der bereit gestellten Informationen zugesichert.

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