Nilpotency of finite groups with Frobenius groups of automorphisms. (Q636847)

The paper under review has the following two main results which are somehow generalizations of previous results of the authors [\textit{E. I. Khukhro}, Bull. Lond. Math. Soc. 40, No. 5, 907-912 (2008; Zbl 1200.17015); Sib. Èlektron. Mat. Izv. 6, 243-250 (2009); \textit{P. Shumyatsky}, Boll. Unione Mat. Ital., Sez. B, Artic. Ric. Mat. (8) 4, No. 3, 731-736 (2001; Zbl 1130.20308); Proc. Am. Math. Soc. 129, No. 12, 3479-3484 (2001; Zbl 0990.20015)]. Theorem 1. Let \(BC\) be a Frobenius group with non-cyclic Abelian kernel \(B\) and with complement \(C\) of order \(t\). Suppose that \(BC\) acts coprimely on a finite group \(G\) in such a manner that \(C_G(C)\) is Abelian and \([C_G(u),\underbrace{C_G(v),\dots,C_G(v)}_{k\text{ copies}}]=1\) for any \(u,v\in B\setminus\{1\}\). Then \(G\) is nilpotent of class bounded by a function depending on \(k\) and \(t\) only. Theorem 2. Let \(BC\) be a Frobenius group with Abelian kernel \(B\) of rank at least three and with complement \(C\) of order \(t\). Suppose that \(BC\) acts coprimely on a finite group \(G\) in such a manner that \(C_G(C)\) is Abelian and \(C_G(b)\) is nilpotent of class at most \(c\) for every \(b\in B\setminus\{1\}\). Then \(G\) is nilpotent of class bounded by a function depending on \(c\) and \(t\) only. The proofs are based on results on graded Lie rings with few non-trivial components, and with many commuting components.