Nonsoluble and non-p-soluble length of finite groups

Abstract

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 minimum number of nonsoluble factors in a 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.