Characterizations of some classes of finite σ-soluble PσT-groups

2017/04/08 by Skiba, Alexander N.
#20D10 #20D15 #20D30 #FOS: Mathematics #Group Theory (math.GR)

paper · doi:10.48550/arxiv.1704.02509

Abstract

Let σ=\σ_i | i∈ I\ be some partition of the set of all primes ℙ and G a finite group. G is said to be σ-soluble if every chief factor H/K of G is a σ_i-group for some i=i(H/K). A set \cal H of subgroups of G is said to be a complete Hall σ-set of G if every member ≠ 1 of \cal H is a Hall σ_i-subgroup of G for some σ_i∈ σ and \cal H contains exactly one Hall σ_i-subgroup of G for every i ∈ I such that σ_i∩ π(G)≠ ∅. A subgroup A of G is said to be σ-permutable in G if G has a complete Hall σ-set \cal H such that AH^x=H^xA for all x∈ G and all H∈ \cal H. We obtain characterizations of finite σ-soluble groups G in which σ-permutability is a transitive relation in G.

Related

• On finite PσT-groups
• On generalized σ-soluble groups
• A Robinson characterization of finite PσT-groups
• On weakly sigma-permutable subgroups of finite groups
• On the lattice of the σ-permutable subgroups of a finite group