- Identification of irreducible groups
- Compatibility with other data libraries
- Loading and unloading recognition data manually

This chapter describes some functions which, given an irreducible matrix group, identify a group in the IRREDSOL library which is conjugate to that group, see Section Identification of irreducible groups. Moreover, Section Compatibility with other data libraries describes how to translate between groups in the IRREDSOL library and the GAP library of irreducible soluble groups. Section Loading and unloading recognition data manually describes some functions which allow to load and unload the recognition data in the IRREDSOL package manually.

`IsAvailableIdIrreducibleSolubleMatrixGroup(`

`) F`

`IsAvailableIdIrreducibleSolvableMatrixGroup(`

`) F`

This function returns `true`

if `IdIrreducibleSolubleMatrixGroup`

(see
IdIrreducibleSolubleMatrixGroup) will work for the irreducible matrix group `G`, and `false`

otherwise.

`IsAvailableIdAbsolutelyIrreducibleSolubleMatrixGroup(`

`) F`

`IsAvailableIdAbsolutelyIrreducibleSolvableMatrixGroup(`

`) F`

This function returns `true`

if `IdIrreducibleSolubleMatrixGroup`

(see
IdIrreducibleSolubleMatrixGroup) will work for the absolutely irreducible matrix group `G`, and `false`

otherwise.

`IdIrreducibleSolubleMatrixGroup(`

`) A`

`IdIrreducibleSolvableMatrixGroup(`

`) A`

If the matrix group `G` is soluble and irreducible over `F
= FieldOfMatrixGroup( G)`, (see FieldOfMatrixGroup), and a conjugate in

`[`

`]`

such that `IrreducibleSolubleMatrixGroup`

(

gap> G := IrreducibleSolubleMatrixGroup(12, 2, 3, 52)^RandomInvertibleMat(12, GF(2));; # <matrix group of size 2340 with 6 generators> gap> IdIrreducibleSolubleMatrixGroup(G); [ 12, 2, 3, 52 ]

`RecognitionIrreducibleSolubleMatrixGroup(`

`[, `

`[, `

`[,`

`]]]) F`

`RecognitionIrreducibleSolubleMatrixGroupNC(`

`[, `

`[, `

`[,`

`]]]) F`

`RecognitionIrreducibleSolvableMatrixGroup(`

`[, `

`[, `

`[,`

`]]]) F`

`RecognitionIrreducibleSolvableMatrixGroupNC(`

`[, `

`[, `

`[,`

`]]]) F`

Let `G` be an irreducible soluble matrix group over a finite field, and let
`wantmat` and `wantmat` be `true`

or `false`

.
These functions identify a conjugate `H` of `G` group in the library.
They return a record which has the following entries:

`id`

- contains the id of
`H`(and thus of`G`); cf.`IdIrreducibleSolubleMatrixGroup`

(IdIrreducibleSolubleMatrixGroup) `mat`

(present if`wantmat`

is`true`

)-
a matrix
`x`such that`G`^{x}= H `group`

(present if`wantgroup`

is`true`

)- the group
`H` `iso`

(present if`wantiso`

is`true`

)- a group isomorphism from the source of
`RepresentationIsomorphism`

(`G`) to the source of`RepresentationIsomorphism`

(`H`).

`RecognitionIrreducibleSolubleMatrixGroup`

and
`RecognitionIrreducibleSolvableMatrixGroupNC`

are
much slower if
`RecognitionIrreducibleSolubleMatrixGroupNC`

does not check its arguments. If
the group `G` is beyond the scope of the IRREDSOL library (see IsAvailableIdIrreducibleSolubleMatrixGroup),
`RecognitionIrreducibleSolubleMatrixGroupNC`

returns `fail`

, while `RecognitionIrreducibleSolubleMatrixGroup`

raises an error.

gap> G := IrreducibleSolubleMatrixGroup(6, 2, 3, 5) ^ > RandomInvertibleMat(6, GF(4)); <matrix group of size 42 with 3 generators> gap> r := RecognitionIrreducibleSolubleMatrixGroup(G, true, false);; gap> r.id; [ 6, 2, 3, 5 ] gap> G^r.mat = CallFuncList(IrreducibleSolubleMatrixGroup, r.id); true

`IdAbsolutelyIrreducibleSolubleMatrixGroup(`

`) A`

`RecognitionAbsolutelyIrreducibleSolubleMatrixGroup(`

`, `

`, `

`) F`

`RecognitionAbsolutelyIrreducibleSolubleMatrixGroupNC(`

`, `

`,`

`) F`

`IdAbsolutelyIrreducibleSolvableMatrixGroup(`

`) A`

`RecognitionAbsolutelyIrreducibleSolubleMatrixGroup(`

`, `

`, `

`) F`

`RecognitionAbsolutelyIrreducibleSolvableMatrixGroupNC(`

`, `

`,`

`) F`

These functions are no longer available. These functions have been replaced by the
functions
`IdIrreducibleSolubleMatrixGroup`

(IdIrreducibleSolubleMatrixGroup),
`RecognitionIrreducibleSolubleMatrixGroup`

(RecognitionIrreducibleSolubleMatrixGroup), or
`RecognitionIrreducibleSolubleMatrixGroupNC`

(RecognitionIrreducibleSolubleMatrixGroupNC).

Note that the ids returned by the functions for absolutely irreducible groups was a triple `[`

`n`, `d`, `k``]`

, while the replacement functions use ids of the form `[`

`n`, `d`, `d`, `k``]`

, where ` d = 1` in the absolutely irreducible case.

A library of irreducible soluble subgroups of `GL(n, p)`, where `p` is a
prime and `p ^{n} leq255` already exists in GAP, see PrimGrp reference manual primgrp:Irreducible Solvable Matrix Groups. The following functions
allow one to translate between between that library and the IRREDSOL library.

`IdIrreducibleSolubleMatrixGroupIndexMS(`

`, `

`, `

`) F`

This function returns the id (see IdIrreducibleSolubleMatrixGroup) of `G`,
where `G` is `IrreducibleSolubleGroupMS`

(`n`, `p`, `k`) (see PrimGrp reference manual primgrp:IrreducibleSolvableGroupMS) .

gap> IdIrreducibleSolubleMatrixGroupIndexMS(6, 2, 5); [ 6, 2, 2, 4 ] gap> G := IrreducibleSolubleGroupMS(6,2,5); <matrix group of size 27 with 2 generators> gap> H := IrreducibleSolubleMatrixGroup(6, 2, 2, 4); <matrix group of size 27 with 3 generators> gap> G = H; false # groups in the libraries need not be the same gap> r := RecognitionIrreducibleSolubleMatrixGroup(G, true, false);; gap> G^r.mat = H; true

`IndexMSIdIrreducibleSolubleMatrixGroup(`

`, `

`, `

`, `

`) F`

This function returns a triple [`n`, `p`, `l`] such that
`IrreducibleSolubleGroupMS`

(`n`, `p`, `l`) (see PrimGrp reference manual primgrp:IrreducibleSolvableGroupMS) is conjugate to
`IrreducibleSolubleMatrixGroup`

(`n`, `q`, `d`, `k`) (see IrreducibleSolubleMatrixGroup).

gap> IndexMSIdIrreducibleSolubleMatrixGroup(6, 2, 2, 7); [ 6, 2, 13 ] gap> G := IrreducibleSolubleGroupMS(6,2,13); <matrix group of size 63 with 3 generators> gap> H := IrreducibleSolubleMatrixGroup(6, 2, 2, 7); <matrix group of size 63 with 3 generators> gap> G = H; false gap> r := RecognitionIrreducibleSolubleMatrixGroup(G, true, false);; gap> G^r.mat = H; true

The data required by the IRREDSOL library is loaded into GAP's workspace automatically whenever required, but is never unloaded automatically. The functions described in this and the previous section describe how to load and unload this data manually. They are only relevant if timing or conservation of memory is an issue.

`LoadAbsolutelyIrreducibleSolubleGroupFingerprints(`

`, `

`) F`

This function loads the fingerprint data required for the recognition
of absolutely irreducible soluble subgroups of `LoadedAbsolutelyIrreducibleSolubleGroupFingerprints() F`

This function returns a list. Each entry consists of an integer `n` and a set `l`. The set
`l` contains all prime powers `q` such that the recognition data for `GL( n, q)` is currently in
memory.

`UnloadAbsolutelyIrreducibleSolubleGroupFingerprints([n [,q]]) F`

This function can be used to delete recognition data for irreducible groups from the GAP workspace. If no
argument is given, all data will be deleted. If only `n` is given, all data for degree `n` (and any
`q`) will be deleted. If `n` and `q` are given, only the data for `GL(n, q)` will be deleted from the
GAP workspace. Use this function if you run out of GAP workspace. The
data is automatically re-loaded when required.

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IRREDSOL manual

November 2022