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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).

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    IRREDSOL manual
    April 2004