Finite groups of bounded rank with an almost regular automorphism
Recall that a finite group \(G\) has rank \(r\) if every subgroup of \(G\) can be generated by \(r\) elements. \textit{A. Shalev} [Isr. J. Math. 82, No. 1-3, 395-404 (1993; Zbl 0817.20026)] proved that if a finite group \(G\) of rank \(r\) admits an automorphism \(\varphi\) with exactly \(m\) fixed points then \(G\) has a soluble subgroup \(H\) of \((r,m)\)-bounded index (i.e. of index bounded in terms of \(r\) and \(m\)), and if in addition \((|G|,|\varphi|)=1\) then \(H\) can be chosen to have \((r,m)\)-bounded derived length. The reviewer [Sib. Mat. Zh. 37, No. 6, 1407-1412 (1996; Zbl 0882.20014)] improved the latter result by showing that \(H\) can be chosen to have \(r\)-bounded derived length. In the present article the author proves the same kind of result in the general case, thus answering the Shalev-Khukhro question 13.56 from the Kourovka notebook [1995; Zbl 0838.20001]. The main case is that a finite \(p\)-group of rank \(r\) admits an automorphism with exactly \(m\) fixed points. The theory of powerful \(p\)-groups allows one to assume the group to be uniformly powerful. Then a certain Lie ring construction due to \textit{A. Shalev} [loc. cit.] is used to reduce the problem to Lie algebras over \(p\)-adic integers. The main effort is applied to such Lie algebras with automorphisms, for which a similar result is proved. The proof is based on a beautiful method developed by the author earlier [in Adv. Math. 153, No. 2, 391-402 (2000; Zbl 0968.20011)] (where he proved that a finite \(p\)-group with an automorphism of order \(p^n\) having exactly \(p^m\) fixed points has a soluble subgroup of \(m\)-bounded derived length and \((p,n,m)\)-bounded index). Very roughly speaking, the assumption that there is no bound for the derived length leads to a limiting situation in zero characteristic, where \textit{V. A. Kreknin}'s theorem [Sib. Mat. Zh. 8, 715-716 (1967; Zbl 0164.03604)] on solubility of Lie algebras with fixed-point free automorphisms produces a contradiction. At the end of the paper the author poses two further conjectures on finite \(p\)-groups of given rank with automorphisms with given number of fixed points.
scientific article; zbMATH DE number 1760716