Characterizations of finite groups with σ-semiembedded subgroups

Author:

Cao Chenchen1,Guo Wenbin2ORCID

Affiliation:

1. School of Mathematics and Statistics, Ningbo University, Ningbo 315211, P. R. China

2. School of Science, Hainan University, Haikou 570225, P. R. China

Abstract

Let [Formula: see text] be a partition of the set of all primes [Formula: see text], [Formula: see text] a finite group and [Formula: see text]. A set [Formula: see text] of subgroups of [Formula: see text] is said to be a complete Hall[Formula: see text]-set of [Formula: see text] if every non-identity member of [Formula: see text] is a Hall [Formula: see text]-subgroup of [Formula: see text] for some [Formula: see text] and [Formula: see text] contains exactly one Hall [Formula: see text]-subgroup of [Formula: see text] for every [Formula: see text]. A subgroup [Formula: see text] of [Formula: see text] is said to be: [Formula: see text]-permutable in [Formula: see text] if [Formula: see text] has a complete Hall [Formula: see text]-set [Formula: see text] such that [Formula: see text] for all [Formula: see text] and all [Formula: see text]; [Formula: see text]-semipermutable in [Formula: see text] if [Formula: see text] has a complete Hall [Formula: see text]-set [Formula: see text] such that [Formula: see text] for all [Formula: see text] and all [Formula: see text]-group [Formula: see text] with [Formula: see text]. We say that [Formula: see text] is [Formula: see text]-semiembedded in [Formula: see text] if there exists a [Formula: see text]-permutable subgroup [Formula: see text] of [Formula: see text] such that [Formula: see text] is a [Formula: see text]-permutable subgroup of [Formula: see text] and [Formula: see text], where [Formula: see text] denotes the subgroup of [Formula: see text] which is generated by all [Formula: see text]-semipermutable subgroups of [Formula: see text] contained in [Formula: see text]. In this paper, we study the influence of [Formula: see text]-semiembedded subgroups on the structure of finite groups. Some known results are generalized and unified.

Funder

National Natural Science Foundation of China

Natural Science Foundation of Jilin Province

Publisher

World Scientific Pub Co Pte Ltd

Subject

Applied Mathematics,Algebra and Number Theory

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