On classes of finite groups with simple non-abelian chief factors

2017/11/06 by Murashka, V. I.
#FOS: Mathematics #Group Theory (math.GR)

paper · doi:10.48550/arxiv.1711.01686

Abstract

Let \mathfrakJ be a class of non-abelian simple groups and \mathfrakX be a class of groups. A chief factor H/K of a group G is called \mathfrakX-central in G provided (H/K)\rtimes G/C_G(H/K)∈\mathfrakX. We say that G is a \mathfrakJcs-\mathfrakX-group if every chief \mathfrakX-factor of G is \mathfrakX-central and other chief factors of G are simple \mathfrakJ-groups. We use \mathfrakX_\mathfrakJcs to denote the class of all \mathfrakJcs-\mathfrakX-groups. A subgroup U of a group G is called \mathfrakX-maximal in G provided that (a) U∈\mathfrakX, and (b) if U≤ V ≤ G and V∈\mathfrakX, then U = V. In this paper we described the structure of \mathfrakJcs-\mathfrakH-groups for a solubly saturated formation \mathfrakH and all hereditary saturated formations \mathfrakF containing all nilpotent groups such that the \mathfrakF_\mathfrakJcs-hypercenter of G coincides with the intersection of all \mathfrakF_\mathfrakJcs-maximal subgroups of G for every group G.

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