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##
#W grpffmat.gd GAP Library Frank Celler
##
##
#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 operations for matrix groups over finite fields.
##
#############################################################################
##
#C IsFFEMatrixGroup( <G> )
##
## <#GAPDoc Label="IsFFEMatrixGroup">
## <ManSection>
## <Filt Name="IsFFEMatrixGroup" Arg='G' Type='Category'/>
##
## <Description>
## tests whether all matrices in <A>G</A> have finite field element entries.
## </Description>
## </ManSection>
## <#/GAPDoc>
##
DeclareSynonym( "IsFFEMatrixGroup", IsFFECollCollColl and IsMatrixGroup );
#############################################################################
##
#M IsFinite( <ffe-mat-grp> )
##
## *Note:* The following implication only holds if there are no infinite
## dimensional matrices.
##
InstallTrueMethod( IsFinite,
IsFFEMatrixGroup and IsFinitelyGeneratedGroup );
#############################################################################
##
#F NicomorphismFFMatGroupOnFullSpace
##
## <ManSection>
## <Func Name="NicomorphismFFMatGroupOnFullSpace" Arg='obj'/>
##
## <Description>
## Compute the permutation action on the full vector space
## </Description>
## </ManSection>
##
DeclareGlobalFunction( "NicomorphismFFMatGroupOnFullSpace" );
#############################################################################
##
#F ProjectiveActionOnFullSpace( <G>, <F>, <n> )
##
## <#GAPDoc Label="ProjectiveActionOnFullSpace">
## <ManSection>
## <Func Name="ProjectiveActionOnFullSpace" Arg='G, F, n'/>
##
## <Description>
## Let <A>G</A> be a group of <A>n</A> by <A>n</A> matrices over a field
## contained in the finite field <A>F</A>.
## <!-- why is <A>n</A> an argument?-->
## <!-- (it should be read off from the group!)-->
## <Ref Func="ProjectiveActionOnFullSpace"/> returns the image of the
## projective action of <A>G</A> on the full row space
## <M><A>F</A>^{<A>n</A>}</M>.
## </Description>
## </ManSection>
## <#/GAPDoc>
##
DeclareGlobalFunction( "ProjectiveActionOnFullSpace" );
#############################################################################
##
#F ConjugacyClassesOfNaturalGroup
##
## <ManSection>
## <Func Name="ConjugacyClassesOfNaturalGroup" Arg='obj'/>
##
## <Description>
## </Description>
## </ManSection>
##
DeclareGlobalFunction( "ConjugacyClassesOfNaturalGroup" );
#############################################################################
##
#F Phi2( <n> ) . . . . . . . . . . . . Modification of Euler's Phi function
##
## <ManSection>
## <Func Name="Phi2" Arg='n'/>
##
## <Description>
## This is needed for the computation of the class numbers of SL(n,q),
## PSL(n,q), SU(n,q) and PSU(n,q)
## </Description>
## </ManSection>
##
DeclareGlobalFunction("Phi2");
#############################################################################
##
#F NrConjugacyClassesGL( <n>, <q> ) . . . . . . . . Class number for GL(n,q)
#F NrConjugacyClassesGU( <n>, <q> ) . . . . . . . . Class number for GU(n,q)
#F NrConjugacyClassesSL( <n>, <q> ) . . . . . . . . Class number for SL(n,q)
#F NrConjugacyClassesSU( <n>, <q> ) . . . . . . . . Class number for SU(n,q)
#F NrConjugacyClassesPGL( <n>, <q> ) . . . . . . . Class number for PGL(n,q)
#F NrConjugacyClassesPGU( <n>, <q> ) . . . . . . . Class number for PGU(n,q)
#F NrConjugacyClassesPSL( <n>, <q> ) . . . . . . . Class number for PSL(n,q)
#F NrConjugacyClassesPSU( <n>, <q> ) . . . . . . . Class number for PSU(n,q)
#F NrConjugacyClassesSLIsogeneous( <n>, <q>, <f> ) . . for SL(n,q) isogeneous
#F NrConjugacyClassesSUIsogeneous( <n>, <q>, <f> ) . . for SU(n,q) isogeneous
##
## <#GAPDoc Label="NrConjugacyClassesGL">
## <ManSection>
## <Func Name="NrConjugacyClassesGL" Arg='n, q'/>
## <Func Name="NrConjugacyClassesGU" Arg='n, q'/>
## <Func Name="NrConjugacyClassesSL" Arg='n, q'/>
## <Func Name="NrConjugacyClassesSU" Arg='n, q'/>
## <Func Name="NrConjugacyClassesPGL" Arg='n, q'/>
## <Func Name="NrConjugacyClassesPGU" Arg='n, q'/>
## <Func Name="NrConjugacyClassesPSL" Arg='n, q'/>
## <Func Name="NrConjugacyClassesPSU" Arg='n, q'/>
## <Func Name="NrConjugacyClassesSLIsogeneous" Arg='n, q, f'/>
## <Func Name="NrConjugacyClassesSUIsogeneous" Arg='n, q, f'/>
##
## <Description>
## The first of these functions compute for given positive integer <A>n</A>
## and prime power <A>q</A> the number of conjugacy classes in the classical
## groups GL( <A>n</A>, <A>q</A> ), GU( <A>n</A>, <A>q</A> ),
## SL( <A>n</A>, <A>q</A> ), SU( <A>n</A>, <A>q</A> ),
## PGL( <A>n</A>, <A>q</A> ), PGU( <A>n</A>, <A>q</A> ),
## PSL( <A>n</A>, <A>q</A> ), PSL( <A>n</A>, <A>q</A> ), respectively.
## (See also <Ref Attr="ConjugacyClasses" Label="attribute"/> and
## Section <Ref Sect="Classical Groups"/>.)
## <P/>
## For each divisor <A>f</A> of <A>n</A> there is a group of Lie type
## with the same order as SL( <A>n</A>, <A>q</A> ), such that its derived
## subgroup modulo its center is isomorphic to PSL( <A>n</A>, <A>q</A> ).
## The various such groups with fixed <A>n</A> and <A>q</A> are called
## <E>isogeneous</E>.
## (Depending on congruence conditions on <A>q</A> and <A>n</A> several of
## these groups may actually be isomorphic.)
## The function <Ref Func="NrConjugacyClassesSLIsogeneous"/> computes the
## number of conjugacy classes in this group.
## The extreme cases <A>f</A> <M>= 1</M> and <A>f</A> <M>= n</M> lead
## to the groups SL( <A>n</A>, <A>q</A> ) and PGL( <A>n</A>, <A>q</A> ),
## respectively.
## <P/>
## The function <Ref Func="NrConjugacyClassesSUIsogeneous"/> is the
## analogous one for the corresponding unitary groups.
## <P/>
## The formulae for the number of conjugacy classes are taken
## from <Cite Key="Mac81"/>.
## <P/>
## <Example><![CDATA[
## gap> NrConjugacyClassesGL(24,27);
## 22528399544939174406067288580609952
## gap> NrConjugacyClassesPSU(19,17);
## 15052300411163848367708
## gap> NrConjugacyClasses(SL(16,16));
## 1229782938228219920
## ]]></Example>
## </Description>
## </ManSection>
## <#/GAPDoc>
##
DeclareGlobalFunction("NrConjugacyClassesGL");
DeclareGlobalFunction("NrConjugacyClassesGU");
DeclareGlobalFunction("NrConjugacyClassesSL");
DeclareGlobalFunction("NrConjugacyClassesSU");
DeclareGlobalFunction("NrConjugacyClassesPGL");
DeclareGlobalFunction("NrConjugacyClassesPGU");
DeclareGlobalFunction("NrConjugacyClassesPSL");
DeclareGlobalFunction("NrConjugacyClassesPSU");
DeclareGlobalFunction("NrConjugacyClassesSLIsogeneous");
DeclareGlobalFunction("NrConjugacyClassesSUIsogeneous");
#############################################################################
##
#E
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