Finite \(p\)-groups with a fixed-point-free automorphisms of order seven. (Q632297)

By the Higman-Kreknin-Kostrikin theorem, if a nilpotent group \(G\) admits a fixed-point-free automorphism of prime order \(q\), then it is nilpotent of class \(\leq h(q)\). In the present paper, for \(q=7\), under the additional assumption that \(G\) is a finite \(p\)-group generated by two elements of order \(p\), it is proved that \(G\) is nilpotent of class less than 7. Some further information is obtained under the assumption that \(p\equiv 1\bmod 7\), which is always true if \(G\) is non-Abelian. It is known that \(h(7)=12\); thus a better bound is obtained due to the restriction on the number of generators. Reviewer's remarks: The author's avoidance of using Lie rings makes the proof unnecessarily long and cumbersome; plus, by using Lie rings the result can be extended to any two-generator \(p\)-group with a fixed-point-free automorphism of order 7.