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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 in the GAP reference manual), 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 Section Irreducible Solvable Matrix Groups in the GAP reference manual. 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 IrreducibleSolvableGroupMS in the GAP reference manual).

    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 IrreducibleSolvableGroupMS in the GAP reference manual) 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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    IRREDSOL manual
    February 2017