Dear GAP-Forume

Alexander B. Konovalov wrote:

Dear Gap-forum,
thank you very much for all who replied to my previous message concerning
Maximal subgroups of SymmetricGroup(9). I suppose it is worthwhile to note
what was the reason to calculate them. There is a problem 12.80 by
V.I.Suschansky in Kourovka notebook (I translate from Russian edition):
Describe all pairs (n,r) of natural n and r, such that the symmetric group
S_{n} contains maximal subgroup, isomorphic to S_{r}.
I have state a task for a graduate diploma project of my student to find
using GAP as much as possible of such pairs (n,r).
It is easy to obtain that for n<9 this pairs are only (3,2), (4,3), (5,4),
(6,5), (7,6) and (8,7). For n=9 we encountered the problem reported in my
previous message.
As far as I understand, it is obvious that the pair (n,n-1) obviously exist
for every n>2, and the problem is to prove that this is the only possible
case. Whether somebody already deal with this problem ?

Looking through the primitive groups up to degree 50, there seem to be two
further examples, (21,7) and (36,9). In general, you would get
(n(n-1)/2,n) for n odd, coing from the action of S_n on unordered pairs.
(For n even, this action lies in the alternating group, so is not
maximal.)

There must be many other such infinite families of examples coming from
other primitive actions of S_n. How about (165,11) coming from the action
on unordered triples, for example? (I have not checked whether this is
really maximal.)

Derek Holt.