Influence of strongly closed 2-subgroups on the structure of finite groups

TL;DR

This paper explores how the property of strong closure of certain subgroups within Sylow 2-subgroups influences the overall structure of finite groups, particularly regarding 2-nilpotence and supersolvability.

Contribution

It provides new structural results on finite groups assuming strong closure conditions on subgroups of order 2^m and 4 within Sylow 2-subgroups, extending previous work.

Findings

Conditions imply 2-nilpotence of the group

Results on supersolvability under strong closure assumptions

Complementary to prior research by Guo and Wei

Abstract

Let $H\leq K$ be subgroups of a group G. We say that H is strongly closed in K with respect to G if whenever $a^g \in K$ where $a \in H, g \in G,$ then $a^g \in H.$ In this paper, we investigate the structure of a group G under the assumption that every subgroup of order $2^m$ (and 4 if m = 1) of a 2- Sylow subgroup S of G is strongly closed in S with respect to G. Some results related to 2-nilpotence and supersolvability of a group G are obtained. This is a complement to Guo and Wei (J. Group Theory 13 (2010), no. 2, 267-276).