[seqfan] Re: Symmetric groups

[Nikos Apostolakis]

On Sun, Nov 27, 2016 at 4:14 PM Frank Adams-Watters <franktaw at netscape.net>
wrote:

> If n divides m!, does the symmetric group S_m always have a subgroup of
> order n?
>
> If so, a comment should be added to A002034 that a(n) is the genus of the
> smallest symmetric group with a subgroup of order n. If not, where is the
> first exception? (8 in S_4?) Is the sequence so described in the OEIS? If
> not, it should be added.
>

The dihedral group is a subgroup of S_4 (<(1,2,3,4), (2,4)> ) and has order
8.

As noted, GAP can be used to calculate all conjugacy classes of subgroups.
For n = 5, there are no subgroups of order 15, 30, 40.  For n =6 there are
no subgroups of orders 15, 30, 40, 45, 80, 90, 144, 180, 240.

 Nikos