Remarks on finite group admitting an automorphism of prime order (Q1924572)

J. Thompson in 1950 proved his well known theorem that a finite group admitting a fixed-point-free automorphism of prime order is nilpotent. The author removes the fixed-point-free hypothesis and obtains with some conditions a decomposition of the group \(G\) as a product of the Fitting subgroup and the fixed point subgroup: Let \(\pi\) be a set of prime numbers, \(G\) a finite \(\pi\)-nilpotent group with a nilpotent \(\pi\)-Hall subgroup and \(\alpha\) an automorphism of \(G\) of prime order \(r\), \(r\notin\pi\). The main result is to prove that \(G=F(G)C_G(\alpha)\) provided any one of the following occurs: a) \(2\notin\pi\); b) \(2\in\pi\) and \(r\) is not a Fermat prime; c) \(2\in\pi\) and the 2-Sylow subgroups of \(G\) are abelian.