Some Nonsimplicity Criteria for Finite Groups

Abstract

If G is a finite group and P a Sylow p-subgroup of G then we can write

$${H_G}\left( P \right)/{O_P},\left( {{N_G}\left( P \right)} \right) \simeq H = PT$$

((1))

where the p’-group T normalises P and acts faithfully on it. In order to be able to discuss such situations concisely we shall reserve the letter P for a p-group, T for a p’-group of automorphisms of P, and H for the semidirect product PT; and we shall say that H is realised in G if (1) holds. We are interested in the question whether H can be realised in a simple group, or perhaps in a perfect group, particularly in cases when P is abelian. (If the answer is yes, one should, of course, go on to ask for a list of the simple groups in which H is realised; but for odd p we do not have such a list even in the simplest cases.) This is a situation in which transfer says something. For instance, the Hall-Wielandt Theorem implies that if the class of P is less than p and o ^p(H) ≠ H then H cannot be realised in a perfect group. Our theorems at least include cases that go beyond this, that is, where o ^p (H) = H.