A note on multiply transitive permutation groups. II

Abstract

This note is a continuation of [I], and also treats rather formal properties of finite multiply transitive permutation groups with relatively high multiple transitivity. The main aim of this note is, roughly speaking, to show that the following assertion holds: Let G be a nontrivial t-ply transitive permutation group of degree n. If the stabilizer of t points in G has an orbit of length m on the remaining n t points, then t must be bounded by a certain (explicit) function depending only on m. If m = 1, then we obtain that t 2, then we obtain that t < p2, wherep is the smallest prime which is strictly greater than m. This is shown in Section 2 as Theorem 2. In Section 1, we prove Theorem 1, which is an extension of Theorem in [I] and at the same time is an odd prime analogue of a result of T. Oyama ([I 1, Theorem 11). Theorem 1 seems to be of a rather technical nature, but it has some important applications, especially it plays an essential role in our proof of Theorem 2. Another application of Theorem 1 is shown in Section 3, where we give an alternative (shortened) proof for some part of the proof of Main Theorem in [3], i.e., the determination of Zp-ply transitive permutation groups whose stabilizer of 2p points is of order prime to p. (See also Miyamoto [S] and Bannai [2, 31.)