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#############################################################################
##
## This file is part of GAP, a system for computational discrete algebra.
## This file's authors include Bettina Eick.
##
## 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
##
#############################################################################
##
#V InfoSpecPcgs
##
DeclareInfoClass( "InfoSpecPcgs" );
#############################################################################
##
#P IsSpecialPcgs( <obj> )
##
## <#GAPDoc Label="IsSpecialPcgs">
## <ManSection>
## <Prop Name="IsSpecialPcgs" Arg='obj'/>
##
## <Description>
## tests whether <A>obj</A> is a special pcgs.
## </Description>
## </ManSection>
## <#/GAPDoc>
##
DeclareProperty( "IsSpecialPcgs", IsPcgs );
InstallTrueMethod( IsPcgs, IsSpecialPcgs );
#InstallTrueMethod(IsPcgsCentralSeries,IsSpecialPcgs);
InstallTrueMethod(IsPcgsElementaryAbelianSeries,IsSpecialPcgs);
#############################################################################
##
#A SpecialPcgs( <pcgs> )
#A SpecialPcgs( <G> )
##
## <#GAPDoc Label="SpecialPcgs">
## <ManSection>
## <Heading>SpecialPcgs</Heading>
## <Attr Name="SpecialPcgs" Arg='pcgs' Label="for a pcgs"/>
## <Attr Name="SpecialPcgs" Arg='G' Label="for a group"/>
##
## <Description>
## computes a special pcgs for the group defined by <A>pcgs</A> or for
## <A>G</A>.
## </Description>
## </ManSection>
## <#/GAPDoc>
##
## A method for `SpecialPcgs(<G>)' must call `SpecialPcgs(Pcgs(<G>))' (this
## is to avoid accidentally forgetting information.)
DeclareAttribute( "SpecialPcgs", IsPcgs );
#############################################################################
##
#A LGHeads( <pcgs> )
##
## <ManSection>
## <Attr Name="LGHeads" Arg='pcgs'/>
##
## <Description>
## returns the LGHeads of the special pcgs <A>pcgs</A>.
## </Description>
## </ManSection>
##
DeclareAttribute( "LGHeads", IsPcgs );
#############################################################################
##
#A LGTails( <pcgs> )
##
## <ManSection>
## <Attr Name="LGTails" Arg='pcgs'/>
##
## <Description>
## returns the LGTails of the special pcgs <A>pcgs</A>.
## </Description>
## </ManSection>
##
DeclareAttribute( "LGTails", IsPcgs );
#############################################################################
##
#A LGWeights( <pcgs> )
##
## <#GAPDoc Label="LGWeights">
## <ManSection>
## <Attr Name="LGWeights" Arg='pcgs'/>
##
## <Description>
## returns the LGWeights of the special pcgs <A>pcgs</A>.
## </Description>
## </ManSection>
## <#/GAPDoc>
##
DeclareAttribute( "LGWeights", IsPcgs );
#############################################################################
##
#A LGLayers( <pcgs> )
##
## <#GAPDoc Label="LGLayers">
## <ManSection>
## <Attr Name="LGLayers" Arg='pcgs'/>
##
## <Description>
## returns the layers of the special pcgs <A>pcgs</A>.
## </Description>
## </ManSection>
## <#/GAPDoc>
##
DeclareAttribute( "LGLayers", IsPcgs );
#############################################################################
##
#A LGFirst( <pcgs> )
##
## <#GAPDoc Label="LGFirst">
## <ManSection>
## <Attr Name="LGFirst" Arg='pcgs'/>
##
## <Description>
## returns the first indices for each layer of the special pcgs <A>pcgs</A>.
## </Description>
## </ManSection>
## <#/GAPDoc>
##
DeclareAttribute( "LGFirst", IsPcgs );
#############################################################################
##
#A LGLength( <G> )
##
## <#GAPDoc Label="LGLength">
## <ManSection>
## <Attr Name="LGLength" Arg='G'/>
##
## <Description>
## returns the length of the LG-series of the group <A>G</A>,
## if <A>G</A> is solvable, and <K>fail</K> otherwise.
##
## <Example><![CDATA[
## gap> G := SmallGroup( 96, 220 );
## <pc group of size 96 with 6 generators>
## gap> spec := SpecialPcgs( G );
## Pcgs([ f1, f2, f3, f4, f5, f6 ])
## gap> LGWeights(spec);
## [ [ 1, 1, 2 ], [ 1, 1, 2 ], [ 1, 1, 2 ], [ 1, 1, 2 ], [ 1, 1, 3 ],
## [ 1, 2, 2 ] ]
## gap> LGLayers(spec);
## [ 1, 1, 1, 1, 2, 3 ]
## gap> LGFirst(spec);
## [ 1, 5, 6, 7 ]
## gap> LGLength( G );
## 3
## gap> p := SpecialPcgs( Pcgs( SmallGroup( 96, 120 ) ) );
## Pcgs([ f1, f2, f3, f4, f5, f6 ])
## gap> LGWeights(p);
## [ [ 1, 1, 2 ], [ 1, 1, 2 ], [ 1, 1, 2 ], [ 1, 2, 2 ], [ 1, 3, 2 ],
## [ 2, 1, 3 ] ]
## ]]></Example>
## <P/>
## Thus the first group, <C>SmallGroup(96, 220)</C>, has a lower nilpotent
## series of length <M>1</M>; that is, the group is nilpotent.
## It is a direct product of its Sylow subgroups.
## Moreover the Sylow <M>2</M>-subgroup is generated by the elements
## <C>f1, f2, f3, f4, f6</C>,
## and the Sylow <M>3</M>-subgroup is generated by <C>f5</C>.
## The lower <M>2</M>-central series of the Sylow <M>2</M>-subgroup
## has length <M>2</M> and the second subgroup in this series is generated
## by <C>f6</C>.
## <P/>
## The second group, <C>SmallGroup(96, 120)</C>, has a lower nilpotent
## series of length <M>2</M> and hence is not nilpotent.
## The second subgroup in this series is just the Sylow <M>3</M>-subgroup
## and it is generated by <C>f6</C>.
## The subgroup generated by <C>f1</C>, <M>\ldots</M>, <C>f5</C> is a
## Sylow <M>2</M>-subgroup of the group and also a head complement to the
## second head of the group.
## Its lower <M>2</M>-central series has length <M>2</M>.
## <P/>
## In this example the <Ref Attr="FamilyPcgs"/> value of the groups used
## was a special pcgs, but this is not necessarily the case.
## For performance reasons it can be worth to enforce this,
## see <Ref Attr="IsomorphismSpecialPcGroup"/>.
## </Description>
## </ManSection>
## <#/GAPDoc>
##
DeclareAttribute( "LGLength", IsGroup );
#############################################################################
##
#A InducedPcgsWrtSpecialPcgs( <G> )
##
## <#GAPDoc Label="InducedPcgsWrtSpecialPcgs">
## <ManSection>
## <Attr Name="InducedPcgsWrtSpecialPcgs" Arg='G'/>
##
## <Description>
## computes an induced pcgs with respect to the special pcgs of the
## parent of <A>G</A>.
## <P/>
## <Ref Attr="InducedPcgsWrtSpecialPcgs"/> will return a pcgs induced by
## <E>a</E> special pcgs (which might differ from the one you had in mind).
## If you need an induced pcgs compatible with a <E>given</E> special pcgs
## use <Ref Func="InducedPcgs"/> for this special pcgs.
## </Description>
## </ManSection>
## <#/GAPDoc>
##
DeclareAttribute( "InducedPcgsWrtSpecialPcgs", IsGroup );
#############################################################################
##
#A CanonicalPcgsWrtSpecialPcgs( <G> )
##
## <ManSection>
## <Attr Name="CanonicalPcgsWrtSpecialPcgs" Arg='G'/>
##
## <Description>
## </Description>
## </ManSection>
##
DeclareAttribute( "CanonicalPcgsWrtSpecialPcgs", IsGroup );
#############################################################################
##
#P IsInducedPcgsWrtSpecialPcgs( <pcgs> )
##
## <#GAPDoc Label="IsInducedPcgsWrtSpecialPcgs">
## <ManSection>
## <Prop Name="IsInducedPcgsWrtSpecialPcgs" Arg='pcgs'/>
##
## <Description>
## tests whether <A>pcgs</A> is induced with respect to a special pcgs.
## </Description>
## </ManSection>
## <#/GAPDoc>
##
DeclareProperty( "IsInducedPcgsWrtSpecialPcgs", IsPcgs );
InstallTrueMethod( IsPcgs, IsInducedPcgsWrtSpecialPcgs );
#############################################################################
##
#P IsCanonicalPcgsWrtSpecialPcgs( <pcgs> )
##
## <ManSection>
## <Prop Name="IsCanonicalPcgsWrtSpecialPcgs" Arg='pcgs'/>
##
## <Description>
## </Description>
## </ManSection>
##
DeclareProperty( "IsCanonicalPcgsWrtSpecialPcgs", IsPcgs );
InstallTrueMethod( IsPcgs, IsCanonicalPcgsWrtSpecialPcgs );
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