Finite Permutation Groups

Abstract

The major part of my thesis is concerned with the size and structure of Sylow p-subgroups of a primitive permutation group. The results of Theorems 2.2 and 2.3 were suggested by similar results of Jordan, Manning, Waiss, and othera, about elements of order p in a primitive group. The following are the three main results: Theorem 2.1 . If G is a transitive permutation group on a set Ω of degree n, and if P is a Sylow p-subgroup of G for some prime p dividing |G|, then the number of points of Ω fixed by P is less than n ⁄ 2 . Theorem 2.2 . Let G be a primitive permutation group on Ω of degree n = kp, where p is a prime, and such that G does not contain the alternating group A n . Let P be a Sylow p-subgroup of G, and suppose that P has no orbits of length greater thin p. Then P has order p unless |P| = 4 and G is PSL(2,5) permuting the 6 points or the 1-dimensional projective geometry PG(1,5), or |P| = 9 and G is the Mathieu group M 11 in its 3-transitive representation of degree 12. This result is due to L. Scott for the case in which G is not 2-transitive and my contribution is the 2-transitive case. Theorem 2.3 . Let G be a 2-transitive permutation group on Ω of degree n = kp + f, for some prime p, such that G does not contain the alternating group A n . Suppose that p divides |G| and that a Sylow p-subgroup P of G has k orbits of length p and f fixed points in Ω. Then P has order p unless f = 0. As the first application of these results we prove Theorem 7.1 below about 2-transitive groups of degree r 2 + 3r + 3, where r is a prime. This problem arose from a conjecture about transitive groups of prime degree, and work of Peter Neumann and Tom McDonough. Theorem 7.1 . If G is a 2-transitive permutation group on Ω of degree n = r 2 + 3r + 3, where r is a prime greater than 3, and such that r divides |G|, then either G contains the alternating group A n , or r is of the form 2 m - 1, a Mersenne prime, for some odd prime m, and G is such that PSL(3,2 m ) ≤ G ≤ PΓL(3,2 m ). Next we turn to 2-transitive groups of degree p 2 , where p is a prime. In looking at the case whore the Sylow p-subgroups are cyclic, the situation arose in which G had an indecomposable representation of degree less than |P| ⁄ 2 . To deal with this, the next theorem, an extension of a result of Felt, was proved. Theorem 9.2 . Let G be a finite group with a cyclic Sylow p-subgroup P of order p k ≥ p 2 , which is a T.I. set. Suppose that G is not p-soluble. Suppose that G has an indecomposable representation ℒ in a field K of characteristic p of degree d ≤ p k , such that P is not contained in the kernel of ℒ. Then ℒ p is indecomposable, C G (P) = PxZ(G), and d ≥ (p k +1) ⁄ 2 . Finally there are some results about 2-transitive groups of degree p 2 , following on from Wielendt's classification of the simply transitive groups: Theorem 12.3 . If G is a 2-transitive group of degree p 2 and P is a Sylow p-subgroup of G, then either |P| ≥ p 4 and G contains A p 2 , for p ≥ 3, or |P| = p 3 and G ≤ Aff(2,p), (and G has PSL(2,p) as a composition factor), or |P| = 3 3 and G is PΓL(2,8) of degree 9, or |P| = 2 3 and G is S 4 of degree 4, or |P| = p 2 . If G is primitive of degree p k and its Sylow p-subgroups are cyclic, we use Theorem 9.2 to extend results of Neumann and Ito, (Theorem 14.2, and Corollary 14.3).