G-covering subgroup systems for some classes of σ-soluble groups

Abstract

Throughout this paper, all groups are finite and G always denotes a finite group.Let σ={σi|i∈I} be a partition of the set of all primes P. The group G is said to be: σ-primary if G is a σi-group for some i=i(G); σ-nilpotent if G=G1×…×Gn for some σ-primary groups G1,…,Gn; σ-soluble if every chief factor of G is σ-primary; σ-full if G possesses a Hall σi-subgroup for all i such that σi∩π(G)≠∅.A subgroup A of G is said to be σ-permutable in G provided G is σ-full and A permutes with every Hall σi-subgroup H of G, that is, AH=HA for all i; G is said to be a PσT-group if σ-permutability is a transitive relation in G, that is, if K is a σ-permutable subgroup of H and H is a σ-permutable subgroup of G, then K is a σ-permutable subgroup of G.Let F be a class of group. Then a set Σ of subgroups of G is called a G-covering subgroup system for the class F if G∈F whenever Σ⊆F.We prove that: If a set of subgroups Σ of G contains at least one supplement to each maximal subgroup of every Sylow subgroup of G, then Σ is a G-covering subgroup system for the classes of all σ-soluble groups, all σ-nilpotent groups, and all σ-soluble PσT-groups.This result gives positive answers to Questions 19.87 and 19.88 in the Kourovka Notebook and, also, allows us to obtain the following characterization of σ-soluble PσT-groups: G is a σ-soluble PσT-group if and only if each maximal subgroup of every Sylow subgroup of G has a supplement T in G such that T is a σ-soluble PσT-group.