[IRREDSOL] 3 Recognition of matrix groups

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# 3 Recognition of matrix groups

### Sections

1. Identification of irreducible groups
2. Compatibility with other data libraries
3. 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.

## 3.1 Identification of irreducible groups

- `IsAvailableIdIrreducibleSolubleMatrixGroup(`G`) F`
- `IsAvailableIdIrreducibleSolvableMatrixGroup(`G`) F`

  This function returns `true` if `IdIrreducibleSolubleMatrixGroup` (see IdIrreducibleSolubleMatrixGroup) will work for the irreducible matrix group G, and `false` otherwise.

- `IsAvailableIdAbsolutelyIrreducibleSolubleMatrixGroup(`G`) F`
- `IsAvailableIdAbsolutelyIrreducibleSolvableMatrixGroup(`G`) F`

  This function returns `true` if `IdIrreducibleSolubleMatrixGroup` (see IdIrreducibleSolubleMatrixGroup) will work for the absolutely irreducible matrix group G, and `false` otherwise.

- `IdIrreducibleSolubleMatrixGroup(`G`) A`
- `IdIrreducibleSolvableMatrixGroup(`G`) A`

  If the matrix group G is soluble and irreducible over F = `FieldOfMatrixGroup`(G), (see FieldOfMatrixGroup), and a conjugate in GL(n, F) of G belongs to the data base of irreducible soluble groups in IRREDSOL, this function returns a list `[`n, q, d, k`]` such that G is conjugate to `IrreducibleSolubleMatrixGroup`(n, q, d, k) (see 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(`G`[,` wantmat`[,` wantgroup`[,`wantiso`]]]) F`
- `RecognitionIrreducibleSolubleMatrixGroupNC(`G`[,` wantmat`[,` wantgroup`[,`wantiso`]]]) F`
- `RecognitionIrreducibleSolvableMatrixGroup(`G`[,` wantmat`[,` wantgroup`[,`wantiso`]]]) F`
- `RecognitionIrreducibleSolvableMatrixGroupNC(`G`[,` wantmat`[,` wantgroup`[,`wantiso`]]]) 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 Gx = 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).

  Note that in most cases, `RecognitionIrreducibleSolubleMatrixGroup` and `RecognitionIrreducibleSolvableMatrixGroupNC` are much slower if wantmat is set to true.

  `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(`G`) A`
- `RecognitionAbsolutelyIrreducibleSolubleMatrixGroup(`G`,` wantmat`,` wantgroup`) F`
- `RecognitionAbsolutelyIrreducibleSolubleMatrixGroupNC(`G`,` wantmat`,`wantgroup`) F`
- `IdAbsolutelyIrreducibleSolvableMatrixGroup(`G`) A`
- `RecognitionAbsolutelyIrreducibleSolubleMatrixGroup(`G`,` wantmat`,` wantgroup`) F`
- `RecognitionAbsolutelyIrreducibleSolvableMatrixGroupNC(`G`,` wantmat`,`wantgroup`) 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.

  ## 3.2 Compatibility with other data libraries

  A library of irreducible soluble subgroups of GL(n, p), where p is a prime and pn 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(`n`,` p`,` k`) 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(`n`,` q`,` d`,` k`) 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

  ## 3.3 Loading and unloading recognition data manually

  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(`n`,` q`) F` This function loads the fingerprint data required for the recognition of absolutely irreducible soluble subgroups of GL(n, q).

- `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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