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#############################################################################
##
#W irredsol.gd GAP group library Mark Short
#W Burkhard Höfling
##
##
#Y Copyright (C) 1993, Murdoch University, Perth, Australia
##
## This file contains the functions and data for the irreducible solvable
## matrix group library. It contains exactly one member for each of the
## 372 conjugacy classes of irreducible solvable subgroups of $GL(n,p)$
## where $1 < n$, $p$ is a prime, and $p^n < 256$.
##
## By well-known theory, this data also doubles as a library of primitive
## solvable permutation groups of non-prime degree $<256$.
##
## This file contains the data from Mark Short's thesis, plus two groups
## missing from that list, subsequently discovered by Alexander Hulpke.
##
#############################################################################
##
#V IrredSolJSGens[] . . . . . . . . . . . . . . . generators for the groups
##
## <ManSection>
## <Var Name="IrredSolJSGens"/>
##
## <Description>
## <C>IrredSolJSGens[<A>n</A>][<A>p</A>][<A>k</A>]</C> is a generating set
## for the <A>k</A>-th JS-maximal of GL(<A>n</A>,<A>p</A>).
## This generating set is polycyclic, i.e. forms an AG-system for the group.
## A JS-maximal is a maximal irreducible solvable subgroup of
## GL(<A>n</A>,<A>p</A>)
## (for a few exceptional small values of n and p this group isn't maximal).
## Every group in the library is generated with reference to the generating
## set of one of these JS-maximals, called its guardian (a group may be a
## subgroup of several JS-maximals but it only has one guardian).
## </Description>
## </ManSection>
##
#DeclareGlobalVariable("IrredSolJSGens");
#############################################################################
##
#V IrredSolGroupList[] . . . . . . . . . . . . . . description of the groups
##
## <ManSection>
## <Var Name="IrredSolGroupList"/>
##
## <Description>
## <C>IrredSolGroupList[<A>n</A>][<A>p</A>][<A>i</A>]</C> is a list containing the information
## about the <A>i</A>-th group from GL(<A>n</A>,<A>p</A>).
## The groups are ordered with respect to the following criteria:
## 1. Increasing size
## 2. Increasing guardian number
## If two groups have the same size and guardian, they are in no particular
## order.
## <P/>
## The list <C>IrredSolGroupList[<A>n</A>][<A>p</A>][<A>i</A>]</C> contains the following info:
## Position: [1]: the size of the group
## [2]: 0 if group is linearly primitive,
## otherwise its minimal block size
## [3]: the absolute value is the number of the group's guardian,
## i.e. its position in 'IrredSolJSGens[<A>n</A>][<A>p</A>]',
## it's negative iff it equals its guardian
## [4..]: the group's generators in normal form
## (with respect to its guardian's AG-system)
## </Description>
## </ManSection>
##
#DeclareGlobalVariable ("IrredSolGroupList");
#############################################################################
##
#F IrreducibleSolvableGroup( <n>, <p>, <i> )
##
## <#GAPDoc Label="IrreducibleSolvableGroup">
## <ManSection>
## <Func Name="IrreducibleSolvableGroup" Arg='n, p, i'/>
##
## <Description>
## This function is obsolete, because for <A>n</A> <M>= 2</M>,
## <A>p</A> <M>= 13</M>, two groups were missing from the
## underlying database. It has been replaced by the function
## <Ref Func="IrreducibleSolvableGroupMS"/>. Please note that the latter
## function does not guarantee any ordering of the groups in the database.
## However, for values of <A>n</A>, <A>p</A>, and <A>i</A> admissible to
## <Ref Func="IrreducibleSolvableGroup"/>,
## <Ref Func="IrreducibleSolvableGroupMS"/> returns a representative of the
## same conjugacy class of subgroups of GL(<A>n</A>, <A>p</A>) as
## <Ref Func="IrreducibleSolvableGroup"/> did before.
## </Description>
## </ManSection>
## <#/GAPDoc>
##
DeclareGlobalFunction("IrreducibleSolvableGroup");
#############################################################################
##
#F IrreducibleSolvableGroupMS( <n>, <p>, <i> )
##
## <#GAPDoc Label="IrreducibleSolvableGroupMS">
## <ManSection>
## <Func Name="IrreducibleSolvableGroupMS" Arg='n, p, i'/>
##
## <Description>
## This function returns a representative of the <A>i</A>-th conjugacy class
## of irreducible solvable subgroup of GL(<A>n</A>, <A>p</A>),
## where <A>n</A> is an integer <M>> 1</M>, <A>p</A> is a prime,
## and <M><A>p</A>^{<A>n</A>} < 256</M>.
## <P/>
## The numbering of the representatives should be
## considered arbitrary. However, it is guaranteed that the <A>i</A>-th
## group on this list will lie in the same conjugacy class in all future
## versions of &GAP;, unless two (or more) groups on the list are discovered
## to be duplicates,
## in which case <Ref Func="IrreducibleSolvableGroupMS"/> will return
## <K>fail</K> for all but one of the duplicates.
## <P/>
## For values of <A>n</A>, <A>p</A>, and <A>i</A> admissible to
## <Ref Func="IrreducibleSolvableGroup"/>,
## <Ref Func="IrreducibleSolvableGroupMS"/> returns a representative of
## the same conjugacy class of subgroups of GL(<A>n</A>, <A>p</A>) as
## <Ref Func="IrreducibleSolvableGroup"/>.
## Note that it currently adds two more groups (missing from the
## original list by Mark Short) for <A>n</A> <M>= 2</M>,
## <A>p</A> <M>= 13</M>.
## </Description>
## </ManSection>
## <#/GAPDoc>
##
DeclareGlobalFunction("IrreducibleSolvableGroupMS");
#############################################################################
##
#F NumberIrreducibleSolvableGroups( <n>, <p> )
##
## <#GAPDoc Label="NumberIrreducibleSolvableGroups">
## <ManSection>
## <Func Name="NumberIrreducibleSolvableGroups" Arg='n, p'/>
##
## <Description>
## This function returns the number of conjugacy classes of
## irreducible solvable subgroup of
## GL(<A>n</A>, <A>p</A>).
## </Description>
## </ManSection>
## <#/GAPDoc>
##
DeclareGlobalFunction("NumberIrreducibleSolvableGroups");
DeclareSynonym("NrIrreducibleSolvableGroups",NumberIrreducibleSolvableGroups);
#############################################################################
##
#F AllIrreducibleSolvableGroups( <func1>, <val1>, <func2>, <val2>, ... )
##
## <#GAPDoc Label="AllIrreducibleSolvableGroups">
## <ManSection>
## <Func Name="AllIrreducibleSolvableGroups"
## Arg='func1, val1, func2, val2, ...'/>
##
## <Description>
## This function returns a list of conjugacy class representatives <M>G</M>
## of matrix groups over a prime field such that
## <M>f(G) = v</M> or <M>f(G) \in v</M>, for all pairs <M>(f,v)</M> in
## (<A>func1</A>, <A>val1</A>), (<A>func2</A>, <A>val2</A>), <M>\ldots</M>.
## The following possibilities for the functions <M>f</M>
## are particularly efficient, because the values can be read off the
## information in the data base:
## <C>DegreeOfMatrixGroup</C> (or
## <Ref BookName="ref" Func="Dimension"/> or
## <Ref BookName="ref" Func="DimensionOfMatrixGroup"/>) for the
## linear degree,
## <Ref BookName="ref" Func="Characteristic"/> for the field characteristic,
## <Ref BookName="ref" Func="Size"/>, <C>IsPrimitiveMatrixGroup</C>
## (or <C>IsLinearlyPrimitive</C>), and
## <C>MinimalBlockDimension</C>>.
## </Description>
## </ManSection>
## <#/GAPDoc>
##
DeclareGlobalFunction("AllIrreducibleSolvableGroups");
#############################################################################
##
#F OneIrreducibleSolvableGroup( <func1>, <val1>, <func2>, <val2>, ...)
##
## <#GAPDoc Label="OneIrreducibleSolvableGroup">
## <ManSection>
## <Func Name="OneIrreducibleSolvableGroup"
## Arg='func1, val1, func2, val2, ...'/>
##
## <Description>
## This function returns one solvable subgroup <M>G</M> of a
## matrix group over a prime field such that
## <M>f(G) = v</M> or <M>f(G) \in v</M>, for all pairs <M>(f,v)</M> in
## (<A>func1</A>, <A>val1</A>), (<A>func2</A>, <A>val2</A>), <M>\ldots</M>.
## The following possibilities for the functions <M>f</M>
## are particularly efficient, because the values can be read off the
## information in the data base:
## <C>DegreeOfMatrixGroup</C> (or
## <Ref BookName="ref" Func="Dimension"/> or
## <Ref BookName="ref" Func="DimensionOfMatrixGroup"/>) for the
## linear degree,
## <Ref BookName="ref" Func="Characteristic"/> for the field characteristic,
## <Ref BookName="ref" Func="Size"/>, <C>IsPrimitiveMatrixGroup</C>
## (or <C>IsLinearlyPrimitive</C>), and
## <C>MinimalBlockDimension</C>>.
## </Description>
## </ManSection>
## <#/GAPDoc>
##
DeclareGlobalFunction("OneIrreducibleSolvableGroup");
#############################################################################
##
#A DegreeOfMatrixGroup(<G>)
##
## <ManSection>
## <Attr Name="DegreeOfMatrixGroup" Arg='G'/>
##
## <Description>
## This function returns the dimension of the underlying vector space,
## same as <C>DimensionOfMatrixGroup</C>
## </Description>
## </ManSection>
##
if not IsBound(DegreeOfMatrixGroup) then
# DegreeOfMatrixGroup is also declared identically in irredsol, so to
# avoid warnings we only define it if necessary
DeclareSynonymAttr("DegreeOfMatrixGroup", DimensionOfMatrixGroup);
fi;
#############################################################################
##
#A MinimalBlockDimension(<G>)
##
## <ManSection>
## <Attr Name="MinimalBlockDimension" Arg='G'/>
##
## <Description>
## The minimum integer <A>n</A> such that the matrix group has an imprimitivity
## system consisting of <A>n</A>-dimensional subspaces of the underlying vector
## space over <C>FieldOfMatrixGroup(G)</C>
## </Description>
## </ManSection>
##
DeclareAttribute("MinimalBlockDimension", IsMatrixGroup);
#############################################################################
##
#P IsPrimitiveMatrixGroup(<G>)
##
## <ManSection>
## <Prop Name="IsPrimitiveMatrixGroup" Arg='G'/>
##
## <Description>
## <K>true</K> if <A>G</A> is primitive over <C>FieldOfMatrixGroup(G)</C>
## </Description>
## </ManSection>
##
DeclareProperty("IsPrimitiveMatrixGroup", IsMatrixGroup);
DeclareSynonymAttr ("IsLinearlyPrimitive", IsPrimitiveMatrixGroup);
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