On the intersection of the \cal F-maximal subgroups and the generalized \cal F-hypercentre of a finite group

2012/05/11 by Wenbin Guo, Guo, Wenbin, Alexander N. Skiba +1
Mathematics · #FOS: Mathematics #Finite Group Theory Research #Group Theory (math.GR) #math.GR

paper · pdf · doi:10.48550/arxiv.1205.2498

arXiv admin note: text overlap with arXiv:1103.4729

arxiv created 2012/05/11 · openalex publication_date 2012/05/11 · arxiv updated 2012/05/20 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

Let \cal F be a class of groups. A chief factor H/K of a group G is called \emph\cal F-central in G provided (H/K)\rtimes (G/C_G(H/K)) ∈ \cal F. We write Z_π\cal F(G) to denote the product of all normal subgroups of G whose G-chief factors of order divisible by at least one prime in π are \cal F-central. We call Z_π\cal F(G) the π\cal F-hypercentre of G. A subgroup U of a group G is called \cal F-maximal in G provided that (a) U∈ \cal F, and (b) if U≤ V≤ G and V∈ \cal F, then U=V. In this paper we study the properties of the intersection of all \cal F-maximal subgroups of a finite group. In particular, we analyze the condition under which Z_π\cal F(G) coincides with the intersection of all \cal F-maximal subgroups of G.

Citations

• On a question of L.A. Shemetkov concerning the intersection of \cal F-maximal subgroups of finite groups

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