Automorphisms fixing elements of prime order in finite groups (Q1358273)

It is proved that if a finite group \(G\) admits an automorphism \(\varphi\) of order \(n\) fixing all elements of prime order or order 4 in \(H=[G,\varphi]\), then a) the prime divisors of \(n\) are exactly the prime divisors of \(|H|\); b) \(H\) is nilpotent of class bounded in terms of \(n\); c) the exponent of \(H\) divides \(n\); d) \([H,\varphi,\dots,\varphi]=1\), where the number of commutations is bounded in terms of \(n\). Part a) of the Theorem gives a new proof of the known result [\textit{B. Huppert}, Endliche Gruppen I (1967; Zbl 0217.07201), Satz IV. 5.12], which states that if a finite \(p\)-group \(G\) admits an automorphism \(\varphi\) fixing all elements of prime order or order 4, then \(\varphi\) has \(p\)-power order.