[GAP Forum] IsMinimalNonmonomial(AlternatingGroup(5));

[Bill Allombert]

On Sun, May 02, 2021 at 03:10:17PM -0700, Marc Keilberg wrote:
> A non-solvable minimal non-monomial group must be simple.  Equivalently, a
> non-simple minimal nonmonomial group is solvable. For then it has a proper,
> non-trivial, normal, monomial and therefore solvable subgroup, the quotient
> by which is also monomial and so solvable by assumption. So we can test for
> simplicity in the non-solvable case, and we can then see why the method
> just kind of dies with an error. Trying to solve this problem for generic
> simple groups would likely be extremely computationally intensive and
> likely ad hoc depending on the family, as we must potentially check all
> (classes) of its subgroups which is generally not a solved problem; even
> the classification of maximal subgroups isn't completely settled.  While
> the minimal nonmonomial solvable groups are classified, I don't think the
> simple ones are.  Even deciding simplicity may be a chore, which may
> explain why the method also throws an error on a non-solvable non-simple
> group like SmallGroup(120,5) or S_5.

Note that the documentation of the function does not say it only applies
to solvable groups.

Cheers,
Bill.