Finite groups admitting the extensions of the automorphisms of a maximal elementary abelian 2-subgroup

Abstract

1. INTRODUCTION AND STATEMENT OF THE THEOREM In [lo], Higman described the structure of a finite p-group admitting an automorphism which cyclically permutes the subgroups of order p. Such a group is either a direct sum of cyclic groups of the same order, a generalized quaternion group, or a Suzuki 2-group (defined in [lo]). Several authors have considered the structure of a finite p-group whose automorphism group acts transitively on the elements of order p of the group. (See [S] and the references cited in it.) Gaschiitz and Yen [S] have shown that if G is a group of odd order and, if for each prime divisor p of the order of G, the automorphism group of G acts transitively on the elements of G of order p, then G is a cyclic extension of a nilpotent Hall subgroup. Here, we consider a variation of the situations mentioned above. We say that a finite group G has the (extension) property B(p) for the prime p if the automorphisms of some elementary abelian p-subgroup E of G of the largest possible order can be obtained as the restrictions to E of the automorphisms of G. Our object here is to prove the following: THEOREM. Let G be a finite group possessing the property g(2), and let (G#)S E W,(G). (A) If O,(G) # (1) and O(G) = (1 ), then one of