The Influence of m-σ-embedded Subgroups on the Structure of Finite Groups

Abstract

Let 𝜎𝜎 = {𝜎𝜎 𝑖𝑖 |𝑖𝑖 ∈ 𝐼𝐼} be some partition of the set of all primes ℙ and G is a finite group. A group is said to be 𝜎𝜎-primary if it is a finite 𝜎𝜎 𝑖𝑖 -group for some 𝑖𝑖. A subgroup 𝐴𝐴 of𝐺𝐺is said to be 𝜎𝜎-subnormal in 𝐺𝐺 if there is a subgroup chain𝐵𝐵〉 for some modular subgroup 𝑀𝑀 and 𝜎𝜎-permutable subgroup 𝐵𝐵 of 𝐺𝐺. Following this, a subgroup 𝐻𝐻 of G is m-𝜎𝜎-embedded in 𝐺𝐺 if there exist an m-𝜎𝜎 -permutable subgroup S and a 𝜎𝜎 -subnormal subgroup T of 𝐺𝐺 such that 𝐻𝐻𝐺𝐺 = 𝐻𝐻𝐻𝐻 and 𝐻𝐻 ∩ 𝐻𝐻 ≤ 𝑆𝑆 ≤ 𝐻𝐻, where 𝐻𝐻 𝐺𝐺 = ⟨𝐻𝐻 𝑥𝑥 |𝑥𝑥 ∈ 𝐺𝐺⟩ is the normal closure of 𝐻𝐻 in 𝐺𝐺. In this paper, we study the structure of 𝐺𝐺 under the condition that some given subgroups of 𝐺𝐺 are m-𝜎𝜎-embedded in 𝐺𝐺. Some available results are generalized.