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