On Π-permutable subgroups of finite groups

2016/06/10 by Guo, Wenbin, Skiba, A. N.
#FOS: Mathematics #Group Theory (math.GR)

paper · doi:10.48550/arxiv.1606.03197

Abstract

Let σ=\σ_i | i∈ I\ be some partition of the set of all primes ℙ and Π a non-empty subset of the set σ. A set \cal H of subgroups of a finite group G is said to be a complete Hall Π-set of G if every member of \cal H is a Hall σ_i-subgroup of G for some σ_i∈ Π and \cal H contains exact one Hall σ_i-subgroup of G for every σ_i∈ Π such that σ_i∩ π(G)≠∅. A subgroup H of G is called Π-quasinormal or Π-permutable in G if G possesses a complete Hall Π-set \cal H=\H₁, … , H_t \ such that AH_i^x=H_i^xA for any i and all x∈ G. We study the embedding properties of H under the hypothesis that H is Π-permutable in G. Some known results are generalized.

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