On σ-subnormality criteria in finite groups

Abstract

Let σ = { σ i : i ∈ I } be a partition of the set P of all prime numbers. A subgroup H of a finite group G is called σ - subnormal in G if there is a chain of subgroups H = H 0 ⊆ H 1 ⊆ ⋯ ⊆ H n = G where, for every i = 1 , … , n , H i − 1 normal in H i or H i / C o r e H i ( H i − 1 ) is a σ j -group for some j ∈ I . In the special case that σ is the partition of P into sets containing exactly one prime each, the σ -subnormality reduces to the familiar case of subnormality. In this paper some σ -subnormality criteria for subgroups of finite groups are studied.