Finite groups with given σ-embedded and σ-n-embedded subgroups

Zhenfeng Wu (), Chi Zhang () and Jianhong Huang ()
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Zhenfeng Wu: University of Science and Technology of China
Chi Zhang: University of Science and Technology of China
Jianhong Huang: Jiangsu Normal University

Indian Journal of Pure and Applied Mathematics, 2017, vol. 48, issue 3, 429-448

Abstract: Abstract Let G be a finite group and σ = {σ i |i∈I} be a partition of the set of all primes P. A set H of subgroups of G is said to be a complete Hall σ-set of G if every non-identity member of H is a Hall σ i -subgroup of G and H contains exactly one Hall σ i -subgroup of G for every σ i ∈ σ(G). A subgroup H is said to be σ-permutable if G possesses a complete Hall σ-set H such that HA x = A x H for all A ∈ H and all x ∈ G. Let H be a subgroup of G. Then we say that: (1) H is σ-embedded in G if there exists a σ-permutable subgroup T of G such that HT = H σG and H ∩ T ≤ H σG , where H σG is the subgroup of H generated by all those subgroups of H which are σ-permutable in G, and H σG is the σ-permutable closure of H, that is, the intersection of all σ-permutable subgroups of G containing H. (2) H is σ-n-embedded in G if there exists a normal subgroup T of G such that HT = H G and H ∩ T ≤ H σG . In this paper, we study the properties of the new embedding subgroups and use them to determine the structure of finite groups.

Keywords: Finite group; σ-embedded subgroup; σ-n-embedded subgroup; σ-permutable subgroup; supersoluble (search for similar items in EconPapers)
Date: 2017
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DOI: 10.1007/s13226-017-0239-2

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