Positive laws in fixed points of automorphisms of finite groups.
Suppose that a finite group \(G\) admits a Frobenius group of automorphisms \(FH\) with kernel \(F\) and complement \(H\) such that the kernel \(F\) is cyclic and acts fixed-point-freely, that is, \(C_G(F)=1\). It is proved that if \(C_G(H)\) satisfies a positive law of degree \(k\), then \(G\) also satisfies a positive law of degree bounded in terms of \(k\) and \(|FH|\). Here, a positive law is a group law of the form \(u\equiv v\), where \(u\) and \(v\) are group words involving only positive powers of the variables. The proofs in the present paper, many of them by Lie ring methods, combine in quite a non-trivial way some ideas and techniques from the recent paper by \textit{E. Khukhro, N. Yu. Makarenko} and \textit{P. Shumyatsky}, [Frobenius groups of automorphisms and their fixed points, \url{arXiv:1010.0343v1}], where it was proved that the exponent of \(G\) is bounded in terms of the exponent of \(C_G(H)\) and \(|FH|\), and if \(C_G(H)\) is nilpotent, then \(G\) is also nilpotent of nilpotency class bounded in terms of the nilpotency class of \(C_G(H)\) and \(|H|\). Note that a finite group satisfying a positive law of degree \(k\) is an extension of a nilpotent group of \(k\)-bounded class by a group of \(k\)-bounded exponent [\textit{R. G. Burns, O. Macedońska, Yu. Medvedev}, J. Algebra 195, No. 2, 510-525 (1997; Zbl 0886.20022)]; the converse is also true: an extension of a nilpotent group of class \(c\) by a group of exponent \(e\) satisfies a positive law of degree bounded in terms of \(c\) and \(e\) [\textit{A. I. Mal'tsev}, Ivanov. Gos. Ped. Inst., Uchenye Zap., Fiz.-Mat. Nauki 4, 107-111 (1953; Zbl 0087.25501)]. In a broader context, for a finite group \(G\) admitting a Frobenius group of automorphisms \(FH\) with fixed-point-free kernel \(F\), a few other results were recently obtained on the structure of \(G\) in terms of \(C_G(H)\) and \(H\) [e.g., \textit{N. Yu. Makarenko, P. Shumyatsky}, Proc. Am. Math. Soc. 138, No. 10, 3425-3436 (2010; Zbl 1214.20028)], although several questions remain open. It is worth noting that examples show that the assumption that \(F\) be cyclic is essential for bounding the nilpotency class (as well as for the result in the present paper), while some other properties can be transferred from \(C_G(H)\) to \(G\) for any Frobenius group \(FH\) [e.g., \textit{E. I. Khukhro}, Algebra Logic 49, No. 6, 551-560 (2011); translation from Algebra Logika 49, No. 6, 819-833 (2010; Zbl 1223.20016)].
scientific article; zbMATH DE number 5960886
scientific article