Finite \(2\)-groups admitting an automorphism of order \(4\) with few fixed points
Based on Kreknin's Theorem on the solubility of Lie rings with a regular automorphism, \textit{A. Shalev} proved [in J. Algebra 157, No. 1, 271-282 (1993; Zbl 0797.20023)] that if a finite \(p\)-group \(P\) admits an automorphism of order \(p^k\) fixing exactly \(p^m\) points then \(P\) is soluble of derived length bounded in terms of \(p\), \(k\), \(m\) (for short, \((p,k,m)\)-bounded). The reviewer [in Mat. Sb. 184, No. 12, 53-64 (1993)] proved that, moreover, \(P\) has a subgroup of \((p, k, m)\)-bounded index whose derived length is \((p,k)\)-bounded. In the present paper, the author proves a stronger result in the special case \(p^k = 4\): If a finite 2-group \(G\) admits an automorphism of order 4 fixing exactly \(2^m\) points then \(G\) has a subgroup of \(m\)- bounded index whose derived subgroup is nilpotent of class 3. This result is related to \textit{L. G. Kovács'} theorem stating that a nilpotent group with a fixed-point-free automorphism of order 4 has a nilpotent derived subgroup of class 3 (in fact, is even centre-by-metabelian) [Math. Z. 75, 277-294 (1961; Zbl 0097.012)]. The proof exploits the language of group rings. One of the bright ideas is a double use of P. Hall's Theorem on the nilpotency of soluble groups.
scientific article; zbMATH DE number 738955