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##
#W gpprmsya.gd GAP Library Frank Celler
#W Alexander Hulpke
##
##
#Y Copyright (C) 1996, Lehrstuhl D für Mathematik, RWTH Aachen, Germany
#Y (C) 1998 School Math and Comp. Sci., University of St Andrews, Scotland
#Y Copyright (C) 2002 The GAP Group
##
## This file contains the declarations for symmetric and alternating
## permutation groups
##
#############################################################################
##
#P IsNaturalSymmetricGroup( <group> )
#P IsNaturalAlternatingGroup( <group> )
##
## <#GAPDoc Label="IsNaturalSymmetricGroup">
## <ManSection>
## <Prop Name="IsNaturalSymmetricGroup" Arg='group'/>
## <Prop Name="IsNaturalAlternatingGroup" Arg='group'/>
##
## <Description>
## A group is a natural symmetric or alternating group if it is
## a permutation group acting as symmetric or alternating group,
## respectively, on its moved points.
## <P/>
## For groups that are known to be natural symmetric or natural alternating
## groups, very efficient methods for computing membership,
## conjugacy classes, Sylow subgroups etc. are used.
## <P/>
## <Example><![CDATA[
## gap> g:=Group((1,5,7,8,99),(1,99,13,72));;
## gap> IsNaturalSymmetricGroup(g);
## true
## gap> g;
## Sym( [ 1, 5, 7, 8, 13, 72, 99 ] )
## gap> IsNaturalSymmetricGroup( Group( (1,2)(4,5), (1,2,3)(4,5,6) ) );
## false
## ]]></Example>
## </Description>
## </ManSection>
## <#/GAPDoc>
##
DeclareProperty( "IsNaturalSymmetricGroup", IsPermGroup );
DeclareProperty( "IsNaturalAlternatingGroup", IsPermGroup );
#############################################################################
##
#P IsAlternatingGroup( <group> )
##
## <#GAPDoc Label="IsAlternatingGroup">
## <ManSection>
## <Prop Name="IsAlternatingGroup" Arg='group'/>
##
## <Description>
## is <K>true</K> if the group <A>group</A> is isomorphic to a
## alternating group.
## </Description>
## </ManSection>
## <#/GAPDoc>
##
DeclareProperty( "IsAlternatingGroup", IsGroup );
#############################################################################
##
#M IsAlternatingGroup( <nat-alt-grp> )
##
InstallTrueMethod( IsAlternatingGroup, IsNaturalAlternatingGroup );
#############################################################################
##
#P IsSymmetricGroup( <group> )
##
## <#GAPDoc Label="IsSymmetricGroup">
## <ManSection>
## <Prop Name="IsSymmetricGroup" Arg='group'/>
##
## <Description>
## is <K>true</K> if the group <A>group</A> is isomorphic to a
## symmetric group.
## </Description>
## </ManSection>
## <#/GAPDoc>
##
DeclareProperty( "IsSymmetricGroup", IsGroup );
#############################################################################
##
#M IsSymmetricGroup( <nat-sym-grp> )
##
InstallTrueMethod( IsSymmetricGroup, IsNaturalSymmetricGroup );
#############################################################################
##
#A SymmetricParentGroup( <grp> )
##
## <#GAPDoc Label="SymmetricParentGroup">
## <ManSection>
## <Attr Name="SymmetricParentGroup" Arg='grp'/>
##
## <Description>
## For a permutation group <A>grp</A> this function returns the symmetric
## group that moves the same points as <A>grp</A> does.
## <Example><![CDATA[
## gap> SymmetricParentGroup( Group( (1,2), (4,5), (7,8,9) ) );
## Sym( [ 1, 2, 4, 5, 7, 8, 9 ] )
## ]]></Example>
## </Description>
## </ManSection>
## <#/GAPDoc>
##
DeclareAttribute("SymmetricParentGroup",IsPermGroup);
#############################################################################
##
#A AlternatingSubgroup( <grp> )
##
## <ManSection>
## <Attr Name="AlternatingSubgroup" Arg='grp'/>
##
## <Description>
## returns the intersection of <A>grp</A> with the alternating group on the
## points moved by <A>grp</A>.
## </Description>
## </ManSection>
##
DeclareAttribute("AlternatingSubgroup",IsPermGroup);
#############################################################################
##
#A OrbitStabilizingParentGroup( <grp> )
##
## <ManSection>
## <Attr Name="OrbitStabilizingParentGroup" Arg='grp'/>
##
## <Description>
## returns the subgroup of <C>SymmetricParentGroup(<A>grp</A>)</C> which stabilizes
## the orbits of <A>grp</A> setwise. (So it is a direct product of wreath
## products of symmetric groups.) It is a natural supergroup for the
## normalizer.
## </Description>
## </ManSection>
##
DeclareAttribute("OrbitStabilizingParentGroup",IsPermGroup);
#############################################################################
##
#E
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