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#############################################################################
##
#W perf.gd GAP Groups Library Alexander Hulpke
##
#H @(#)$Id: perf.gd,v 4.14 2001/10/06 20:30:23 gap Exp $
##
#Y Copyright (C) 1997, Lehrstuhl D fuer Mathematik, RWTH Aachen, Germany
##
## This file contains the declarations for the Holt/Plesken library of
## perfect groups
##
Revision.perf_gd :=
"@(#)$Id: perf.gd,v 4.14 2001/10/06 20:30:23 gap Exp $";
PERFRec := fail; # indicator that perf0.grp is not loaded
PERFSELECT := [];
PERFGRP := [];
#############################################################################
##
#C IsPerfectLibraryGroup(<G>) identifier for groups constructed from the
## library (used for perm->fp isomorphism)
##
DeclareCategory("IsPerfectLibraryGroup", IsGroup );
#############################################################################
##
#O PerfGrpConst(<filter>,<descriptor>)
##
DeclareConstructor("PerfGrpConst",[IsGroup,IsList]);
#############################################################################
##
#F PerfGrpLoad(<size>) force loading of secondary files, return index
##
DeclareGlobalFunction("PerfGrpLoad");
#############################################################################
##
#A PerfectIdentification(<G>) . . . . . . . . . . . . id. for perfect groups
##
## This attribute is set for all groups obtained from the perfect groups
## library and has the value `[<size>,<nr>]' if the group is obtained with
## these parameters from the library.
##
DeclareAttribute("PerfectIdentification", IsGroup );
#############################################################################
##
#F SizesPerfectGroups()
##
DeclareGlobalFunction("SizesPerfectGroups");
#############################################################################
##
#F NumberPerfectGroups( <size> ) . . . . . . . . . . . . . . . . . . . . . .
##
## returns the number of non-isomorphic perfect groups of size <size> for
## each positive integer <size> up to $10^6$ except for the eight sizes
## listed at the beginning of this section for which the number is not
## yet known. For these values as well as for any argument out of range it
## returns `fail'.
##
DeclareGlobalFunction("NumberPerfectGroups");
DeclareSynonym("NrPerfectGroups",NumberPerfectGroups);
#############################################################################
##
#F NumberPerfectLibraryGroups( <size> ) . . . . . . . . . . . . . . . . . .
##
## returns the number of perfect groups of size <size> which are available
## in the library of finite perfect groups. (The purpose of the function
## is to provide a simple way to formulate a loop over all library groups
## of a given size.)
##
DeclareGlobalFunction("NumberPerfectLibraryGroups");
DeclareSynonym("NrPerfectLibraryGroups",NumberPerfectLibraryGroups);
#############################################################################
##
#F PerfectGroup( [<filt>, ]<size>[, <n>] )
#F PerfectGroup( [<filt>, ]<sizenumberpair> )
##
## returns a group which is isomorphic to the library group specified
## by the size number `[ <size>, <n> ]' or by the two separate
## arguments <size> and <n>, assuming a default value of $<n> = 1$.
## The optional argument <filt> defines the filter in which the group is
## returned.
## Possible filters so far are `IsPermGroup' and `IsSubgroupFpGroup'.
## In the latter case, the generators and relators used coincide with those
## given in~\cite{HP89}.
##
DeclareGlobalFunction("PerfectGroup");
#############################################################################
##
#F DisplayInformationPerfectGroups( <size> ) . . . . . . . . . . . . . . . .
#F DisplayInformationPerfectGroups( <size>, <n> ) . . . . . . . . . . . . .
#F DisplayInformationPerfectGroups( [ <size>, <n> ] ) . . . . . . . . . . .
##
## `DisplayInformationPerfectGroups' displays some invariants of the <n>-th
## group of order <size> from the perfect groups library.
##
## If no value of <n> has been specified, the invariants will be displayed
## for all groups of size <size> available in the library.
## The information provided for $G$ includes the following items:
## \beginlist%unordered
## \item{$\bullet$}
## a headline containing the size number `[ <size>, <n> ]' of $G$
## in the form `<size>.<n>' (the suffix `.<n>' will be suppressed
## if, up to isomorphism, $G$ is the only perfect group of order
## <size>),
## \item{$\bullet$} a message if $G$ is simple or quasisimple, i.e.,
## if the factor group of $G$ by its centre is simple,
## \item{$\bullet$} the ``description'' of the structure of $G$ as it is
## given by Holt and Plesken in~\cite{HP89} (see below),
## \item{$\bullet$} the size of the centre of $G$ (suppressed, if $G$ is
## simple),
## \item{$\bullet$} the prime decomposition of the size of $G$,
## \item{$\bullet$} orbit sizes for a faithful permutation representation
## of $G$ which is provided by the library (see below),
## \item{$\bullet$} a reference to each occurrence of $G$ in the tables of
## section 5.3 of \cite{HP89}. Each of these references
## consists of a class number and an internal number $(i,j)$ under which
## $G$ is listed in that class. For some groups, there is more than one
## reference because these groups belong to more than one of the classes
## in the book.
## \endlist
##
DeclareGlobalFunction("DisplayInformationPerfectGroups");
#############################################################################
##
#F SizeNumbersPerfectGroups( <factor1>, <factor2>, ... )
##
## `SizeNumbersPerfectGroups' returns a list of pairs, each entry consisting
## of a group order and the number of those groups in the library of perfect
## groups that contain the specified factors <factor1>, <factor2>, ...
## among their composition factors.
##
## Each argument must either be the name of a simple group or an integer
## which stands for the product of the sizes of one or more cyclic factors.
## (In fact, the function replaces all integers among the arguments
## by their product.)
##
## The following text strings are accepted as simple group names.
## \beginlist%unordered
## \item{$\bullet$} `A<n>' or `A(<n>)' for the alternating groups $A_n$,
## $5\leq n\leq9$, for example `A5' or `A(6)'.
## \item{$\bullet$} `L<n>(<q>)' or `L(<n>,<q>)' for $PSL(n,q)$, where
## $n\in\{2,3\}$ and $q$ a prime power, ranging
## \itemitem{$\circ$}%unordered
## for $n=2$ from 4 to 125
## \itemitem{$\circ$} for $n=3$ from 2 to 5
## \item{$\bullet$} `U<n>(<q>)' or `U(<n>,<q>)' for $PSU(n,q)$, where
## $n\in\{3,4\}$ and $q$ a prime power, ranging
## \itemitem{$\circ$}%unordered
## for $n=3$ from 3 to 5
## \itemitem{$\circ$} for $n=4$ from 2 to 2
## \item{$\bullet$} `Sp4(4)' or `S(4,4)' for the symplectic group $S(4,4)$,
## \item{$\bullet$} `Sz(8)' for the Suzuki group $Sz(8)$,
## \item{$\bullet$} `M<n>' or `M(<n>)' for the Mathieu groups $M_{11}$,
## $M_{12}$, and $M_{22}$, and
## \item{$\bullet$} `J<n>' or `J(<n>)' for the Janko groups $J_1$ and
## $J_2$.
## \endlist
##
## Note that, for most of the groups, the preceding list offers two
## different names in order to be consistent with the notation used in
## \cite{HP89} as well as with the notation used in the
## `DisplayCompositionSeries' command of {\GAP}. However, as the names are
## compared as text strings, you are restricted to the above choice. Even
## expressions like `L2(2^5)' are not accepted.
##
## As the use of the term $PSU(n,q)$ is not unique in the literature, we
## mention that in this library it denotes the factor group of $SU(n,q)$ by
## its centre, where $SU(n,q)$ is the group of all $n \times n$ unitary
## matrices with entries in $GF(q^2)$ and determinant 1.
##
## The purpose of the function is to provide a simple way to formulate a
## loop over all library groups which contain certain composition factors.
##
DeclareGlobalFunction("SizeNumbersPerfectGroups");
#############################################################################
##
#E
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