Primitive permutation groups containing a p-element of small degree, p a prime, II

Abstract

Received December 5, 1974 In this paper we consider primitive permutation groups which contain an element of prime order p with at most 2p - 2 cycles of length p. We classify those groups whose orders are divisible byp3. Our result is an extension of the result in [ 111, and extends a theorem of O’Nan similar to the one used in [l 11. As we mentioned in [l 11, these results extend early work of Jordan, Manning and others (see [16, 13.10; and 51). We shall prove the following theorem. THEOREM. S u pp ose that G is a primitive permutation group on a set Q of II points such that for some prime p dividing 1 G 1, G contains an element of order p with Y cycles of length p, where r < 2p - 2. Then one of the following is true. (i) p3 does not divide / G 1. (ii) G is the alternating or symmetric group A, or S, in its natural representation on n points; 1 < Y < [n/p]. (iii) ASL(2,p) < G < AGL(2,p), G permutes the p2 points of the afine plane, and r is p or p - 1. (iv) PSL(3, p) < G < PPL(3, p); G permutes the 1 + p + p2 points or hyperplanes of the projective plane, and r is p OY p + 1. (v) p is 3 and either G is PPL(2, 8) p ermuting the nine points of the projective line, r is 2 or 3, or G is the Mathieugroup M,, of degree 12, r is 3 or 4. (vi) p is 2, Y = 2, and G is PGL(2, 5) permuting the six points of the projective line or G is PSL(2, 7) of d e g ree 7; (note that PSL(2, 7) ‘v GL(3,2) which appears in (iv)) or G is AGL(3,2), p ermuting the eight points of the a&e space. Remarks. (0.1) Most of the notation used follows the conventions of Wielandt’s book [16]. W e a 1 wa y s assume that G is a transitive permutation group on Q. If R is a subgroup of G which permutes a set A, we denote by 278