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By Kantor W.M.

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It is the aim to show that transitive on conjugates of g E G, H n Hg H G is doubly and, for every has a normal complement in H Then a theorem of Suzuki [2J comes into action and yields the conclusion of the theorem above. The just mentioned proposition allows to prove doubly transitivity at a rather early stage, and it provides one with sufficiently many subgroups to which induction hypothesis can be applied. FINITE GROUPS 24 REFERENCES :1] D. Gorenstein, Finite groups; Harper and New York 1968.

Some related results Surprisingly, the fact that p is prime is not important for the proof of Theorem 1, and a better result may be formulated. Definition 3 Let group of degree positive integer.

Let G be a finite group. Denote by O(G) maximal subgroup of odd order. Denote by the minimal normal subgroup of G/N N G the 0' (G) for which has odd order. THEOREM. abelian Let G S2-subgroups. direct product of ~ be ~ finite group with Then 0' (G/O(G)) is the 2-group and simple groups of the following types: n 2 ) , (1) PSL(2, (2) PSL(2, q) q > 3 n > , q - 3 , 43 1 or 5 (mod 8) , 44 FINITE GROUPS (3 ) ~ simple group involution J where K or (mod 8) 5 of M such that - -for - -each -- M is isomorphic to PSL(2, q) , q - 3 Simple groups of the third type have not yet been completely determined.

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