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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

klin@cs.bgu.ac.il

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