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GAP 4.8.9 installation with standard packages -- copy to your CoCalc project to get it
Project: cocalc-sagemath-dev-slelievre
Views: 418346#############################################################################
##
#W perf.gd GAP Groups Library Alexander Hulpke
##
##
#Y Copyright (C) 1997, Lehrstuhl D für Mathematik, RWTH Aachen, Germany
##
## This file contains the declarations for the Holt/Plesken library of
## perfect groups
##
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)
##
## <ManSection>
## <Filt Name="IsPerfectLibraryGroup" Arg='G' Type='Category'/>
##
## <Description>
## </Description>
## </ManSection>
##
DeclareCategory("IsPerfectLibraryGroup", IsGroup );
#############################################################################
##
#O PerfGrpConst(<filter>,<descriptor>)
##
## <ManSection>
## <Oper Name="PerfGrpConst" Arg='filter,descriptor'/>
##
## <Description>
## </Description>
## </ManSection>
##
DeclareConstructor("PerfGrpConst",[IsGroup,IsList]);
#############################################################################
##
#F PerfGrpLoad(<size>) force loading of secondary files, return index
##
## <ManSection>
## <Func Name="PerfGrpLoad" Arg='size'/>
##
## <Description>
## </Description>
## </ManSection>
##
DeclareGlobalFunction("PerfGrpLoad");
#############################################################################
##
#A PerfectIdentification(<G>) . . . . . . . . . . . . id. for perfect groups
##
## <#GAPDoc Label="PerfectIdentification">
## <ManSection>
## <Attr Name="PerfectIdentification" Arg='G'/>
##
## <Description>
## This attribute is set for all groups obtained from the perfect groups
## library and has the value <C>[<A>size</A>,<A>nr</A>]</C> if the group is obtained with
## these parameters from the library.
## </Description>
## </ManSection>
## <#/GAPDoc>
##
DeclareAttribute("PerfectIdentification", IsGroup );
#############################################################################
##
#F SizesPerfectGroups()
##
## <#GAPDoc Label="SizesPerfectGroups">
## <ManSection>
## <Func Name="SizesPerfectGroups" Arg=''/>
##
## <Description>
## This is the ordered list of all numbers up to <M>10^6</M> that occur as
## sizes of perfect groups.
## One can iterate over the perfect groups library with:
## <Example><![CDATA[
## gap> for n in SizesPerfectGroups() do
## > for k in [1..NrPerfectLibraryGroups(n)] do
## > pg := PerfectGroup(n,k);
## > od;
## > od;
## ]]></Example>
## </Description>
## </ManSection>
## <#/GAPDoc>
##
DeclareGlobalFunction("SizesPerfectGroups");
#############################################################################
##
#F NumberPerfectGroups( <size> ) . . . . . . . . . . . . . . . . . . . . . .
##
## <#GAPDoc Label="NumberPerfectGroups">
## <ManSection>
## <Func Name="NumberPerfectGroups" Arg='size'/>
##
## <Description>
## returns the number of non-isomorphic perfect groups of size <A>size</A> for
## each positive integer <A>size</A> up to <M>10^6</M> 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 <K>fail</K>.
## </Description>
## </ManSection>
## <#/GAPDoc>
##
DeclareGlobalFunction("NumberPerfectGroups");
DeclareSynonym("NrPerfectGroups",NumberPerfectGroups);
#############################################################################
##
#F NumberPerfectLibraryGroups( <size> ) . . . . . . . . . . . . . . . . . .
##
## <#GAPDoc Label="NumberPerfectLibraryGroups">
## <ManSection>
## <Func Name="NumberPerfectLibraryGroups" Arg='size'/>
##
## <Description>
## returns the number of perfect groups of size <A>size</A> 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.)
## </Description>
## </ManSection>
## <#/GAPDoc>
##
DeclareGlobalFunction("NumberPerfectLibraryGroups");
DeclareSynonym("NrPerfectLibraryGroups",NumberPerfectLibraryGroups);
#############################################################################
##
#F PerfectGroup( [<filt>, ]<size>[, <n>] )
#F PerfectGroup( [<filt>, ]<sizenumberpair> )
##
## <#GAPDoc Label="PerfectGroup">
## <ManSection>
## <Heading>PerfectGroup</Heading>
## <Func Name="PerfectGroup" Arg='[filt, ]size[, n]'
## Label="for group order (and index)"/>
## <Func Name="PerfectGroup" Arg='[filt, ]sizenumberpair'
## Label="for a pair [ order, index ]"/>
##
## <Description>
## returns a group which is isomorphic to the library group specified
## by the size number <C>[ <A>size</A>, <A>n</A> ]</C> or by the two
## separate arguments <A>size</A> and <A>n</A>, assuming a default value of
## <M><A>n</A> = 1</M>.
## The optional argument <A>filt</A> defines the filter in which the group is
## returned.
## Possible filters so far are <Ref Func="IsPermGroup"/> and
## <Ref Func="IsSubgroupFpGroup"/>.
## In the latter case, the generators and relators used coincide with those
## given in <Cite Key="HP89"/>.
## <Example><![CDATA[
## gap> G := PerfectGroup(IsPermGroup,6048,1);
## U3(3)
## gap> G:=PerfectGroup(IsPermGroup,823080,2);
## A5 2^1 19^2 C 19^1
## gap> NrMovedPoints(G);
## 6859
## ]]></Example>
## </Description>
## </ManSection>
## <#/GAPDoc>
##
DeclareGlobalFunction("PerfectGroup");
#############################################################################
##
#F DisplayInformationPerfectGroups( <size>[, <n>] ) . . . . . . . . . . . .
#F DisplayInformationPerfectGroups( <sizenumberpair>] ) . . . . . . . . . .
##
## <#GAPDoc Label="DisplayInformationPerfectGroups">
## <ManSection>
## <Heading>DisplayInformationPerfectGroups</Heading>
## <Func Name="DisplayInformationPerfectGroups" Arg='size[, n]'
## Label="for group order (and index)"/>
## <Func Name="DisplayInformationPerfectGroups" Arg='sizenumberpair'
## Label="for a pair [ order, index ]"/>
##
## <Description>
## <Ref Func="DisplayInformationPerfectGroups" Label="for group order (and index)"/>
## displays some invariants of the <A>n</A>-th group of order <A>size</A>
## from the perfect groups library.
## <P/>
## If no value of <A>n</A> has been specified, the invariants will be
## displayed for all groups of size <A>size</A> available in the library.
## <P/>
## Alternatively, also a list of length two may be entered as the only
## argument, with entries <A>size</A> and <A>n</A>.
## <P/>
## The information provided for <M>G</M> includes the following items:
## <List>
## <Item>
## a headline containing the size number <C>[ <A>size</A>, <A>n</A> ]</C> of <M>G</M>
## in the form <C><A>size</A>.<A>n</A></C> (the suffix <C>.<A>n</A></C> will be suppressed
## if, up to isomorphism, <M>G</M> is the only perfect group of order
## <A>size</A>),
## </Item>
## <Item>
## a message if <M>G</M> is simple or quasisimple, i.e.,
## if the factor group of <M>G</M> by its centre is simple,
## </Item>
## <Item>
## the <Q>description</Q> of the structure of <M>G</M> as it is
## given by Holt and Plesken in <Cite Key="HP89"/> (see below),
## </Item>
## <Item>
## the size of the centre of <M>G</M> (suppressed, if <M>G</M> is
## simple),
## </Item>
## <Item>
## the prime decomposition of the size of <M>G</M>,
## </Item>
## <Item>
## orbit sizes for a faithful permutation representation
## of <M>G</M> which is provided by the library (see below),
## </Item>
## <Item>
## a reference to each occurrence of <M>G</M> in the tables of
## section 5.3 of <Cite Key="HP89"/>. Each of these references
## consists of a class number and an internal number <M>(i,j)</M> under which
## <M>G</M> 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.
## </Item>
## </List>
## <Example><![CDATA[
## gap> DisplayInformationPerfectGroups( 30720, 3 );
## #I Perfect group 30720: A5 ( 2^4 E N 2^1 E 2^4 ) A
## #I size = 2^11*3*5 orbit size = 240
## #I Holt-Plesken class 1 (9,3)
## gap> DisplayInformationPerfectGroups( 30720, 6 );
## #I Perfect group 30720: A5 ( 2^4 x 2^4 ) C N 2^1
## #I centre = 2 size = 2^11*3*5 orbit size = 384
## #I Holt-Plesken class 1 (9,6)
## gap> DisplayInformationPerfectGroups( Factorial( 8 ) / 2 );
## #I Perfect group 20160.1: A5 x L3(2) 2^1
## #I centre = 2 size = 2^6*3^2*5*7 orbit sizes = 5 + 16
## #I Holt-Plesken class 31 (1,1) (occurs also in class 32)
## #I Perfect group 20160.2: A5 2^1 x L3(2)
## #I centre = 2 size = 2^6*3^2*5*7 orbit sizes = 7 + 24
## #I Holt-Plesken class 31 (1,2) (occurs also in class 32)
## #I Perfect group 20160.3: ( A5 x L3(2) ) 2^1
## #I centre = 2 size = 2^6*3^2*5*7 orbit size = 192
## #I Holt-Plesken class 31 (1,3)
## #I Perfect group 20160.4: simple group A8
## #I size = 2^6*3^2*5*7 orbit size = 8
## #I Holt-Plesken class 26 (0,1)
## #I Perfect group 20160.5: simple group L3(4)
## #I size = 2^6*3^2*5*7 orbit size = 21
## #I Holt-Plesken class 27 (0,1)
## ]]></Example>
## </Description>
## </ManSection>
## <#/GAPDoc>
##
DeclareGlobalFunction("DisplayInformationPerfectGroups");
#############################################################################
##
#F SizeNumbersPerfectGroups( <factor1>, <factor2>, ... )
##
## <#GAPDoc Label="SizeNumbersPerfectGroups">
## <ManSection>
## <Func Name="SizeNumbersPerfectGroups" Arg='factor1, factor2, ...'/>
##
## <Description>
## <Ref Func="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
## <A>factor1</A>, <A>factor2</A>, ...
## among their composition factors.
## <P/>
## 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.)
## <P/>
## The following text strings are accepted as simple group names.
## <List>
## <Item>
## <C>A<A>n</A></C> or <C>A(<A>n</A>)</C> for the alternating groups
## <M>A_{<A>n</A>}</M>,
## <M>5 \leq n \leq 9</M>, for example <C>A5</C> or <C>A(6)</C>.
## </Item>
## <Item>
## <C>L<A>n</A>(<A>q</A>)</C> or <C>L(<A>n</A>,<A>q</A>)</C> for
## PSL<M>(n,q)</M>, where
## <M>n \in \{ 2, 3 \}</M> and <M>q</M> a prime power, ranging
## <List>
## <Item>
## for <M>n = 2</M> from 4 to 125
## </Item>
## <Item>
## for <M>n = 3</M> from 2 to 5
## </Item>
## </List>
## </Item>
## <Item>
## <C>U<A>n</A>(<A>q</A>)</C> or <C>U(<A>n</A>,<A>q</A>)</C> for
## PSU<M>(n,q)</M>, where
## <M>n \in \{ 3, 4 \}</M> and <M>q</M> a prime power, ranging
## <List>
## <Item>
## for <M>n = 3</M> from 3 to 5
## </Item>
## <Item>
## for <M>n = 4</M> from 2 to 2
## </Item>
## </List>
## </Item>
## <Item>
## <C>Sp4(4)</C> or <C>S(4,4)</C> for the symplectic group Sp<M>(4,4)</M>,
## </Item>
## <Item>
## <C>Sz(8)</C> for the Suzuki group Sz<M>(8)</M>,
## </Item>
## <Item>
## <C>M<A>n</A></C> or <C>M(<A>n</A>)</C> for the Mathieu groups
## <M>M_{11}</M>, <M>M_{12}</M>, and <M>M_{22}</M>, and
## </Item>
## <Item>
## <C>J<A>n</A></C> or <C>J(<A>n</A>)</C> for the Janko groups
## <M>J_1</M> and <M>J_2</M>.
## </Item>
## </List>
## <P/>
## 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 Key="HP89"/> as well as with the notation used in the
## <Ref Func="DisplayCompositionSeries"/> command of &GAP;.
## However, as the names are
## compared as text strings, you are restricted to the above choice. Even
## expressions like <C>L2(2^5)</C> are not accepted.
## <P/>
## As the use of the term PSU<M>(n,q)</M> is not unique in the literature,
## we mention that in this library it denotes the factor group of
## SU<M>(n,q)</M> by its centre, where SU<M>(n,q)</M> is the group of all
## <M>n \times n</M> unitary matrices with entries in <M>GF(q^2)</M>
## and determinant 1.
## <P/>
## The purpose of the function is to provide a simple way to formulate a
## loop over all library groups which contain certain composition factors.
## </Description>
## </ManSection>
## <#/GAPDoc>
##
DeclareGlobalFunction("SizeNumbersPerfectGroups");
#############################################################################
##
#E