Finite \(p\)-groups admitting an automorphism of order \(p\) with a small number of fixed points
It is well-known that the nilpotent length of a nilpotent group having a regular automorphism of prime order \(p\) cannot exceed a number \(h=h(p)\) [cf. \textit{G. Higman}, J. Lond. Math. Soc. 32, 321-334 (1957; Zbl 0079.03203)]. The first result of the paper reviewed asserts that a finite \(p\)-group having an automorphism of order \(p\) with \(p^ m\) fixed points contains a subgroup of index \(p^{m(mp(h(p)+m+1)+1)}\) and nilpotency length \(\leq h(p)+1\). The second result asserts that if in addition the above \(p\)-group is \(s\)-step solvable, then it contains a subgroup of index \(p^{m(mp((p-1)^ s-1)/(p-2)+m+1)+1)}\), whose nilpotency length is bounded by \(((p-1)^ s-1)/(p-2)+1\). An example is given, to show that there exist no function \(k(p,m)\) and fixed constant \(c\) (not depending on \(p\)), such that any finite metabelian \(p\)-group admitting an automorphism of order \(p\) with \(p^ m\) fixed points has a subgroup of index \(k(p,m)\) of nilpotency length \(\leq c\). This solves in the negative the question 8.81 a) of the Kourovka Notebook [8th. ed., Novosibirsk (1982; Zbl 0509.20001)].
scientific article; zbMATH DE number 3964157
scientific article