> < ^ From:

> < ^ Subject:

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.

> < [top]