[Top] [Up] [Previous] [Next] [Index]

# 3 Recognition of matrix groups

### Sections

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

This chapter describes some functions which, given an irreducible matrix group, identify a group in the IRREDSOL library which is conjugate to that group.

## 3.1 Identification of irreducible groups

- `IdIrreducibleSolvableMatrixGroupAvailable(`G`) F`

  This function returns true if `IdIrreducibleSolvableMatrixGroup` (see IdIrreducibleSolvableMatrixGroup will work for the group G.

- `IdIrreducibleSolvableMatrixGroup(`G`) A`

  If the matrix group G is solvable and irreducible over F = `FieldOfMatrixGroup`(G), (see FieldOfMatrixGroup), and a conjugate in GL(n, F) of G belongs to the database of irreducible solvable groups in IRREDSOL, this function returns a list `[`n`,` q`,` d`,` k`]` such that G is conjugate to `IrreducibleSolvableMatrixGroup`(n, q, d, k) (see IrreducibleSolvableMatrixGroup).

  ## 3.2 Identification of absolutely irreducible groups

- `IdAbsolutelyIrreducibleSolvableMatrixGroupAvailable(`G`) F`

  This function returns true if `IdAbsolutelyIrreducibleSolvableMatrixGroup` (see IdAbsolutelyIrreducibleSolvableMatrixGroup will work for the group G.

- `IdAbsolutelyIrreducibleSolvableMatrixGroup(`G`) A`

  If the matrix group G is solvable and absolutely irreducible, and if a conjugate in GL(n, F) of G belongs to the database of irreducible solvable groups in IRREDSOL, this function returns a list `[`n`,` q`,` k`]` such that G is conjugate to `AbsolutelyIrreducibleSolvableMatrixGroup`(n, q, k) (see AbsolutelyIrreducibleSolvableMatrixGroup).

- `RecognitionAbsolutelyIrreducibleSolvableMatrixGroup(`G`,` wantmat`,` wantgroup`) F`
- `RecognitionAbsolutelyIrreducibleSolvableMatrixGroupNC(`G`,` wantmat`,`wantgroup`) F`

  Let G be an absolutely irreducible solvable matrix group over a finite field. Theses 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. `IdAbsolutelyIrreducibleSolvableMatrixGroup` (IdAbsolutelyIrreducibleSolvableMatrixGroup

  `mat` (optional)
    a matrix x such that Gx = H

  `group` (optional)
    the group H

  The entries `mat` and `group` are only present if the booleans wantmat and/or wantgroup are true, respectively. Note that in most cases, the function may be much slower if wantmat is set to true.

  The `NC` version does not check its arguments. It returns `fail` if the group G is beyond the scope of the IRREDSOL library; see `IdAbsolutelyIrreducibleSolvableMatrixGroupAvailable` (IdAbsolutelyIrreducibleSolvableMatrixGroupAvailable), while the ordinary version raises an error in this case.

  ## 3.3 Compatibility with other data libraries

  A library of irreducible solvable subgroups of GL(n, p), where p is a prime and pn leq 255 already exists in GAP. The following functions allow to translate between these libraries.

- `IdIrreducibleSolvableMatrixGroupIndexMS(`n`,` p`,` k`) F`

  This function returns the id (see IdIrreducibleSolvableMatrixGroup) of G, where G is `IrreducibleSolvableGroupMS`(n, p, k) (see IrreducibleSolvableGroupMS).

- `IndexMSIdIrreducibleSolvableMatrixGroup(`n`,` q`,` d`,` k`) F`

  This function returns a triple [n, p, l] such that `IrreducibleSolvableGroupMS`(n, p, l)' (see IrreducibleSolvableGroupMS) is conjugate to `IrreducibleSolvableMatrixGroup`(n, q, d, k) (see IrreducibleSolvableMatrixGroup).

  [Top] [Up] [Previous] [Next] [Index]