# Nonsoluble and non-$p$-soluble length of finite groups

Khukhro, E. I. and Shumyatsky, P. (2013) Nonsoluble and non-$p$-soluble length of finite groups. [MIMS Preprint]

Every finite group $G$ has a normal series each of whose factors either is soluble or is a direct product of nonabelian simple groups. We define the nonsoluble length $\lambda (G)$ as the number of nonsoluble factors in a shortest series of this kind. Upper bounds for $\lambda (G)$ appear in the study of various problems on finite, residually finite, and profinite groups. We prove that $\lambda (G)$ is bounded in terms of the maximum $2$-length of soluble subgroups of $G$, and that $\lambda (G)$ is bounded by the maximum Fitting height of soluble subgroups. For an odd prime $p$, the non-$p$-soluble length $\lambda _p(G)$ is introduced, and it is proved that $\lambda _p(G)$ does not exceed the maximum $p$-length of $p$-soluble subgroups. We conjecture that for a given prime $p$ and a given proper group variety ${\frak V}$ the non-$p$-soluble length $\lambda _p(G)$ of finite groups $G$ whose Sylow $p$-subgroups belong to ${\frak V}$ is bounded. In this paper we prove this conjecture for any variety that is a product of several soluble varieties and varieties of finite exponent. As an application of the results obtained, an error is corrected in the proof of the main result of the second author's paper Multilinear commutators in residually finite groups'', \emph{Israel J. Math.} \textbf{189} (2012), 207--224.
Item Type: MIMS Preprint finite groups, Fitting height, $p$-length, nonsoluble length MSC 2010, the AMS's Mathematics Subject Classification > 20 Group theory and generalizations Professor Evgeny Khukhro 10 Dec 2013 08 Nov 2017 18:18 http://eprints.maths.manchester.ac.uk/id/eprint/2080