> < ^ Date: Thu, 13 Apr 2000 16:38:18 +0200 (IST)
< ^ From: Mikhail Klin <klin@cs.bgu.ac.il >
^ Subject: question 3

On 11.04 2000 Alexander Konovalov asked:
> 3. And the last related problem: whether somebody already have checked that
> S_165 contains the maximal subgroup isomorphic to S_11, which is coming from
> the action on unordered triples ?

Dear Alexander Konovalov,
Regarding your third question to forum,
let me once more to explain
(in more strict terms).
The question when symmetric group of degree n
in its action on the set of all m-element subsets
of the n-element set is maximal in the symmetric
(alternating) group of degree {n \choose m}
is completely solved, see exactly the references
in my previous message.
In addition to these references
I can advise you to use a book
by E.Bannai and T.Ito "Algebraic Combinatorics",
translated into Russian by "Mir", in 1987.
Section 2.1.1 of the Supplement 2
gives a short clear review of this subject.

If you like to neglect this theoretical fact
and to use GAP in order to confirm
that S_11 is a maximal subgroup of A_165,
then you may do this as follows.
1. Construct the corresponding action H of degree 165.
2. Describe all 6 non-trivial H-invariant graphs.
3 Use GRAPE to show that all these graphs have the same
automorphism group isomorphic to H.
4. Use simple consequences of CFSG to show
that each 2-transitive overgroup of H includes A_165.

Hope this helps.

Best regards,
Mikhail Klin

Please, find here my present address:
Dr. Mikhail KLin
Department of Mathematics and Computer Science
Ben-Gurion University of the Negev
P.O.Box 653, Beer-Sheva 84105, Israel
Tel: (0)7/6477-802 (office: note new phone from 03.99)
(0)7/641-37-15 (home: note new phone from 28.10.98)
e-mail: klin@indigo.cs.bgu.ac.il

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