turkmath.org

This talk describes joint results with P. Shumyatsky (University of Brasilia). Every nite group $G$ has a normal series each of whose factors either is soluble or is a direct product of nonabelian simple groups. The nonsoluble length $\lambda(G)$ is dened as the number of nonsoluble factors in a shortest series of this kind. Upper bounds for $\lambda(G)$ are needed in the study of various problems on nite, residually nite, and pronite groups. It is proved 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 in similar fashion, and it is proved that $\lambda_p(G)$ does not exceed the maximum $p$-length of $p$-soluble subgroups. It is conjectured that for a given prime $p$ and a given proper group variety $V$ the non-$p$-soluble length $\lambda_p(G)$ of nite groups $G$ whose Sylow $p$-subgroups belong to $V$ is bounded. This conjecture is proved for any variety that is a product of several soluble varieties and varieties of nite exponent.