On almost regular automorphisms of prime order (Q1204785)

Suppose that a periodic nilpotent group \(G\) admits an automorphism \(\varphi\) of prime order \(p\) having exactly \(m\) fixed points. Higman's Theorem states that if \(m = 1\) then the nilpotency class of \(G\) is bounded by ``Higman's function'' \(h(p)\). The reviewer proved [in Mat. Sb. 181, No. 9, 1207-1219 (1990; Zbl 0713.17013)] that, for \(m\) finite, \(G\) has a subgroup of index bounded in terms of \(p\) and \(m\), which is nilpotent of class bounded by a function depending on \(p\) only. The author improves the bound for the nilpotency class up to the best possible value \(h(p)\) at the expense of taking the subgroup generated by all \(s(p,m)\)th powers, for some \(s(p,m)\) depending on \(p\) and \(m\) only, instead of taking a subgroup of bounded index. (Note, however, that if \(G\) is a \(p\)-group, then such a subgroup has also bounded index.) The proof is based on using the technique of the associated Lie rings and the results mentioned above.