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#############################################################################
##
#W grpffmat.gd GAP Library Frank Celler
##
#H @(#)$Id: grpffmat.gd,v 4.18 2002/04/15 10:04:44 sal Exp $
##
#Y Copyright (C) 1996, Lehrstuhl D fuer 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.
##
Revision.grpffmat_gd :=
"@(#)$Id: grpffmat.gd,v 4.18 2002/04/15 10:04:44 sal Exp $";
#############################################################################
##
#C IsFFEMatrixGroup
##
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 NicomorphismOfFFEMatrixGroup
##
DeclareGlobalFunction( "NicomorphismOfFFEMatrixGroup" );
#############################################################################
##
#F ProjectiveActionOnFullSpace( <G>, <F>, <n> )
##
## Let <G> be a group of <n> by <n> matrices over a field contained in the
## finite field <F>.
#T why is <n> an argument?
#T (it should be read off from the group!)
## `ProjectiveActionOnFullSpace' returns the image of the projective action
## of <G> on the full row space $<F>^<n>$.
##
DeclareGlobalFunction( "ProjectiveActionOnFullSpace" );
#############################################################################
##
#F ConjugacyClassesOfNaturalGroup
##
DeclareGlobalFunction( "ConjugacyClassesOfNaturalGroup" );
#############################################################################
##
#F Phi2( <n> ) . . . . . . . . . . . . Modification of Euler's Phi function
##
## This is needed for the computation of the class numbers of SL(n,q),
## PSL(n,q), SU(n,q) and PSU(n,q)
##
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
##
## The first of these functions compute for given $<n> \in N$ and prime
## power $<q>$ the number of conjugacy classes in the classical groups
## $GL( <n>, <q> )$, $GU( <n>, <q> )$, $SL( <n>, <q> )$, $SU( <n>, <q>
## )$, $PGL( <n>, <q> )$, $PGU( <n>, <q> )$, $PSL( <n>, <q> )$, $PSL(
## <n>, <q> )$, respectively. (See also "ConjugacyClasses!attribute" and
## Section~"Classical Groups".)
##
## For each divisor $<f>$ of $<n>$ there is a group of Lie type
## with the same order as $SL( <n>, <q> )$, such that its derived
## subgroup modulo its center is isomorphic to $PSL( <n>, <q> )$.
## The various such groups with fixed $<n>$ and $<q>$ are called
## *isogeneous*. (Depending on congruence conditions on $<q>$ and $<n>$
## several of these groups may actually be isomorphic.) The function
## `NrConjugacyClassesSLIsogeneous' computes the number of conjugacy
## classes in this group. The extreme cases $<f> = 1$ and $<f> = n$ lead
## to the groups $SL( <n>, <q> )$ and $PGL( <n>, <q> )$, respectively.
##
## The function `NrConjugacyClassesSUIsogeneous' is the analogous one for
## the corresponding unitary groups.
##
## The formulae for the number of conjugacy classes are taken
## from~\cite{Mac81}.
##
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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