On almost regular automorphisms of finite \(p\)-groups (Q1580859)

This remarkable paper contains a positive solution of the last major open problem in the theory of almost regular \(p\)-automorphisms of finite \(p\)-groups (based on the corresponding result on Lie rings). Suppose that a finite \(p\)-group \(P\) admits an automorphism \(\varphi\) of order \(p^k\) having exactly \(p^m\) fixed points. What restrictions on the structure of \(P\) can be obtained in terms of \(p^k\) and \(p^m\)? Though a \(p\)-automorphism of a finite \(p\)-group cannot be regular, i.~e. without non-trivial fixed points, it is theorems of G.~Higman and V.~A.~Kreknin on regular automorphisms of Lie rings that are instrumental in this kind of problems. In the case of \(|\varphi|=p\), \textit{J.~L.~Alperin} [Proc. Am. Math. Soc. 13, 175-180 (1962; Zbl 0104.02801)] used Higman's theorem and the associated Lie rings to show that the derived length of \(P\) is bounded in terms of \(p\) and \(m\) (for short, ``\((p,m)\)-bounded''). \textit{E.~I.~Khukhro} [Mat. Zametki 38, No. 5, 652-657 (1985; Zbl 0598.20024)] proved that, moreover, \(P\) then has a subgroup of \((p,m)\)-bounded index which is nilpotent of \(p\)-bounded class. In the general case of \(|\varphi|=p^k\), \textit{A.~Shalev} [J. Algebra 157, No. 1, 271-282 (1993; Zbl 0797.20023)] used Kreknin's theorem and the ``\(p\)-adic'' construction of a Lie ring for uniformly powerful \(p\)-groups to show that the derived length of \(P\) is \((p,k,m)\)-bounded. Using the Mal'cev correspondence as well as powerful \(p\)-groups, \textit{E.~I.~Khukhro} [Mat. Sb. 184, No. 12, 53-64 (1993; Zbl 0836.20018)] proved that, moreover, \(P\) has a subgroup of \((p,k,m)\)-bounded index which is soluble of \(p^k\)-bounded derived length. Results on almost regular \(p\)-automorphisms of finite \(p\)-groups are related to the theory of \(p\)-groups of maximal class and of given coclass (C.~R.~Leedham-Green, S.~McKay, A.~Mann, M.~Newman, S.~Donkin, A.~Shalev, E.~I.~Zelmanov and others). The theory of \(p\)-groups of maximal class is virtually equivalent to the above situation with \(|\varphi|=p\) and \(|C_P(\varphi)|=p\). A large part of the theory of \(p\)-groups of given coclass, the uncovered case, almost amounts to the above situation with \(|\varphi|=p^k\) and \(|C_P(\varphi)|=p\). The strong restriction \(|C_P(\varphi)|=p\) implies a very strong conclusion, the existence of a subgroup of bounded index which is nilpotent of class \(2\). This enabled the reviewer to conjecture that there always exists a subgroup of \((p,k,m)\)-bounded index which is soluble of \(m\)-bounded derived length (even nilpotent of \(m\)-bounded class for \(|\varphi|=p\)). This conjecture is independent of the above-mentioned results of Alperin-Khukhro and Shalev-Khukhro, in which the derived length (nilpotency class) of a subgroup is bounded in terms of the order of the automorphism. \textit{Yu.~Medvedev} [J. Lond. Math. Soc., II. Ser. 58, No. 1, 27-37 (1998; Zbl 0951.20010)] reduced this conjecture to the analogous one on automorphisms of Lie rings whose additive groups are finite \(p\)-groups. (In the case of \(|\varphi|=p\) such a reduction also follows from Khukhro's theorem via Lazard's correspondence for \(p\)-groups of class \(\leq p-1\).) Then \textit{Yu.~Medvedev} [(*) J. Lond. Math. Soc., II. Ser. 59, No. 3, 787-798 (1999; Zbl 0940.17001)] proved the conjecture in the special case of \(|\varphi|=p\). In the paper under review Khukhro's conjecture is proved in the general case, thus solving in the positive Problem 14.96 from the ``Kourovka Notebook'' (1999; Zbl 0943.20003). Due to Medvedev's reduction the proof is essentially about Lie rings. Roughly speaking, the author passes to a certain saturated limiting situation in ``zero characteristic'', where Kreknin's theorem produces a non-trivial Abelian ideal. Then either the ``\(\varphi\)-dimension'' of the Lie ring diminishes and induction can be applied, or the Lie ring becomes virtually nilpotent of class 2. The method of proof can be viewed as an ingenious development of the earlier methods of \textit{A.~Shalev} [Invent. Math. 115, No. 2, 315-345 (1994; Zbl 0795.20009)] and \textit{A.~Shalev} and \textit{E.~I.~Zelmanov} [Math. Proc. Camb. Philos. Soc. 111, No. 3, 417-421 (1992; Zbl 0813.20030)], where the Lie rings were equipped with the action of a transcendental variable \(\vartheta\) reflecting taking \(p\)-th powers in the group, and of \textit{Yu.~Medvedev} [(*), loc. cit.], where a new Lie ring multiplication was defined by lifting the old one to certain \(p^d\)-th roots. Although involving an implicit passage to the limiting object, the method of the author produces quite effective a bound for the derived length of a subring of \((p,m,k)\)-bounded index: such a subring \(N\) can be chosen so that \(\gamma_3(\gamma_3(\dots\gamma_3(N)\dots))=0\), where \(\gamma_3\) appears \(2^m-1\) times. The author conjectures that here \(2^m-1\) could be reduced to \(m\). He also mentions that the methods of this paper are used in his forthcoming paper proving Shalev's conjecture that a finite \(p\)-group of rank \(r\) with an automorphism having \(p^m\) fixed points has a subgroup of \((p,m,r)\)-bounded index and \(r\)-bounded derived length (independently of the order of the automorphism), thus solving in the positive Problem 13.56 in the ``Kourovka Notebook''.