Finite Groups Admitting a Fixed-Point-Free Automorphism of Order 4

Abstract

Recently, in a remarkable piece of work [4, 5] John Thompson has proved a result which implies as an immediate corollary the well-known Frobenius conjecture, namely that a finite group admitting a fixed-point-free automorphism (i. e., leaving only the identity element fixed) of prime order must be nilpotent. However, non-nilpotent groups are known which admit fixedpoint-free automorphisms of composite order. In all these cases one notices that the groups in question are solvable. Although the sample is rather restricted, it is not too unnatural to ask whether the condition that a finite group admit such an automorphism is strong enough to force solvability of the group. This question is related to another problem, which seems. equally difficult, which asks whether a finite group containing a cyclic subgroup which is its own normalizer must be composite. In the present paper we shall prove that a group G possessiilg a fixedpoint-f ree automorphism of order 4 is solvable. Although many of the ideas used carry over to the case in which 4 has order pq, and especially 2q, our key lemmas use the fact that 4 has order 4 in a crucial way. The proof depends upon a theorem of Philip Ilall which asserts that a finite group G is solvable if for every factorization of o(G) into relatively prime numbers m and n, G contains a subgroup of order m. We show (Lemma 7) that a group G which has a fixed-point-free automorphism of order 4 satisfies the conditions of H all's theorem. Once we k-now that G is solvable it is not difficult to prove that its commutator subgroup is nilpotent (Theorem 2). This -fact was also observed by Thompson. Graham Higman has shown [3] that there is a bound to the class of a p-group P which possesses an automorphism p of prime order q without fixedpoints. This does not carry over to automorphisms of composite order, for at the end of the paper we give an example due to Thompson of a family of p-groups of arbitrary high class each of which admits a fixed-point-free automorphism of order 4.