Groups with maximal subgroups of Sylow subgroups σ-permutably embedded

Abstract

Abstract Let σ = { σ i : i ∈ I } ${\sigma=\{\sigma_{i}:i\in I\}}$ be some partition of the set of all primes ℙ ${\mathbb{P}}$ , G a finite group and σ ⁢ ( G ) = { σ i : σ i ∩ π ⁢ ( G ) ≠ ∅ } ${\sigma(G)=\{\sigma_{i}:\sigma_{i}\cap\pi(G)\neq\emptyset\}}$ . A set 1 ∈ ℋ ${1\in{\mathcal{H}}}$ of subgroups of G is said to be a complete Hall σ-set of G if every non-identity group in ℋ ${{\mathcal{H}}}$ is a Hall σ i ${\sigma_{i}}$ -subgroup of G for some σ i ∈ σ ⁢ ( G ) ${\sigma_{i}\in\sigma(G)}$ and ℋ ${{\mathcal{H}}}$ contains exactly one Hall σ i ${\sigma_{i}}$ -subgroup of G for every σ i ∈ σ ⁢ ( G ) ${\sigma_{i}\in\sigma(G)}$ . A subgroup H of G is called σ-permutable (resp. σ-permutably embedded) in G if G possesses a complete Hall σ-set ℋ = { 1 , H 1 , … , H t } ${{\mathcal{H}}=\{1,H_{1},\ldots,H_{t}\}}$ such that A ⁢ H i x = H i x ⁢ A ${AH_{i}^{x}=H_{i}^{x}A}$ for any i and all x ∈ G ${x\in G}$ (resp. if H has a complete Hall σ-set and every Hall σ i ${\sigma_{i}}$ -subgroup of H is also a Hall σ i ${\sigma_{i}}$ -subgroup of some σ-permutable subgroup of G). In this paper, we classify the finite groups G such that either every maximal subgroup of every Sylow subgroup of G is σ-permutable in G or every maximal subgroup of every Sylow subgroup of G is σ-permutably embedded in G.