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#############################################################################
##
## 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
##
## This file contains the functions that calculate ordinary and rational
## classes for permutation groups.
##
#############################################################################
##
#M Enumerator( <xorb> ) . . . . . . . . . for conj. classes in perm. groups
##
## The only difference to the enumerator for external orbits is a better
## `Position' (and `PositionCanonical') method.
##
BindGlobal( "NumberElement_ConjugacyClassPermGroup", function( enum, elm )
local xorb, G, rep;
xorb := UnderlyingCollection( enum );
G := ActingDomain( xorb );
rep := RepOpElmTuplesPermGroup( true, G, [ elm ],
[ Representative( xorb ) ],
TrivialSubgroup( G ), StabilizerOfExternalSet( xorb ) );
if rep = fail then
return fail;
else
return PositionCanonical( enum!.rightTransversal, rep ^ -1 );
fi;
end );
InstallMethod( Enumerator,
[ IsConjugacyClassPermGroupRep ],
xorb -> EnumeratorByFunctions( xorb, rec(
NumberElement := NumberElement_ConjugacyClassPermGroup,
ElementNumber := ElementNumber_ExternalOrbitByStabilizer,
rightTransversal := RightTransversal( ActingDomain( xorb ),
StabilizerOfExternalSet( xorb ) ) ) ) );
#############################################################################
##
#M <cl1> = <cl2> . . . . . . . . . . . . . . . . . . . for conjugacy classes
##
InstallMethod( \=,"classes for perm group", IsIdenticalObj,
[ IsConjugacyClassPermGroupRep, IsConjugacyClassPermGroupRep ],
function( cl1, cl2 )
if not IsIdenticalObj( ActingDomain( cl1 ), ActingDomain( cl2 ) ) then
TryNextMethod();
fi;
return RepOpElmTuplesPermGroup( true, ActingDomain( cl1 ),
[ Representative( cl1 ) ],
[ Representative( cl2 ) ],
StabilizerOfExternalSet( cl1 ),
StabilizerOfExternalSet( cl2 ) ) <> fail;
end );
#############################################################################
##
#M <g> in <cl> . . . . . . . . . . . . . . . . . . . . for conjugacy classes
##
InstallMethod( \in,"perm class rep", IsElmsColls,
[ IsPerm, IsConjugacyClassPermGroupRep ],
function( g, cl )
local G,c;
if CycleStructurePerm(g)<>CycleStructurePerm(Representative(cl)) then
return false;
fi;
if HasAsList(cl) or HasAsSSortedList(cl) then
TryNextMethod();
fi;
G := ActingDomain( cl );
if AttemptPermRadicalMethod(G,"CENT") and g in G then
# use TF method
c:=TFCanonicalClassRepresentative(G,[g,Representative(cl)]:conjugacytest,useradical:=false);
return c<>fail and c[1][2]=c[2][2];
else
return RepOpElmTuplesPermGroup( true, ActingDomain( cl ),
[ g ], [ Representative( cl ) ],
TrivialSubgroup( G ),
StabilizerOfExternalSet( cl ) ) <> fail;
fi;
end );
#############################################################################
##
#M Enumerator( <rcl> ) . . . . . . . . . of rational class in a perm. group
##
## The only difference to the enumerator for rational classes is a better
## `Position' (and `PositionCanonical') method.
##
BindGlobal( "NumberElement_RationalClassPermGroup", function( enum, elm )
local rcl, G, rep, gal, T, pow, t;
rcl := UnderlyingCollection( enum );
G := ActingDomain( rcl );
rep := Representative( rcl );
gal := RightTransversalInParent( GaloisGroup( rcl ) );
T := enum!.rightTransversal;
for pow in [ 1 .. Length( gal ) ] do
# if gal[pow]=0 then the rep is the identity , no need to worry.
t := RepOpElmTuplesPermGroup( true, G,
[ elm ], [ rep ^ Int( gal[ pow ] ) ],
TrivialSubgroup( G ),
StabilizerOfExternalSet( rcl ) );
if t <> fail then
break;
fi;
od;
if t = fail then
return fail;
else
return ( pow - 1 ) * Length( T ) + PositionCanonical( T, t ^ -1 );
fi;
end );
InstallMethod( Enumerator,
[ IsRationalClassPermGroupRep ],
rcl -> EnumeratorByFunctions( rcl, rec(
NumberElement := NumberElement_RationalClassPermGroup,
ElementNumber := ElementNumber_RationalClassGroup,
rightTransversal := RightTransversal( ActingDomain( rcl ),
StabilizerOfExternalSet( rcl ) ) ) ) );
InstallOtherMethod( CentralizerOp, [ IsRationalClassGroupRep ],
StabilizerOfExternalSet );
#############################################################################
##
#M <cl1> = <cl2> . . . . . . . . . . . . . . . . . . . for rational classes
##
InstallMethod( \=, IsIdenticalObj, [ IsRationalClassPermGroupRep,
IsRationalClassPermGroupRep ],
function( cl1, cl2 )
if ActingDomain( cl1 ) <> ActingDomain( cl2 ) then
TryNextMethod();
fi;
# the Galois group of the identity is <0>, therefore we have to do this
# extra test.
return Order(Representative(cl1))=Order(Representative(cl2)) and
ForAny( RightTransversalInParent( GaloisGroup( cl1 ) ), e ->
RepOpElmTuplesPermGroup( true, ActingDomain( cl1 ),
[ Representative( cl1 ) ],
[ Representative( cl2 ) ^ Int( e ) ],
StabilizerOfExternalSet( cl1 ),
StabilizerOfExternalSet( cl2 ) ) <> fail );
end );
#############################################################################
##
#M <g> in <cl> . . . . . . . . . . . . . . . . . . . . for rational classes
##
InstallMethod( \in, true, [ IsPerm, IsRationalClassPermGroupRep ], 0,
function( g, cl )
# the Galois group of the identity is <0>, therefore we have to do this
# extra test.
return Order(Representative(cl))=Order(g) and
ForAny(DecomposedRationalClass(cl),x->g in x);
end );
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