A Proof That Finite Groups with a Fixed-Point-Free Automorphism of Prime Order Are Nilpotent

Abstract

A theorem of Frobenius [5] states that if G is a finite group, H is a proper subgroup, of G which is its own normalizer and has trivial intersection with each of its conjugates, then the set of elements of G which do not lie in any conjugate of H , together with the identity element, forms a normal subgroup N of G , and an element h € H , h ≠ 1 , induces an automorphism of N which leaves only the identity element fixed. Conversely, if a group N possesses a fixed-point-free automorphism σ of prime order, then the holomorph (split extension) of N by {σ} is a group G with {σ} in the role of H . Hence, groups N which can arise in Frobenius’ theorem are precisely those groups with fixed-point-free automorphisms of prime order [4] .