Finite Groups with σ-Subnormal Schmidt Subgroups

Abstract

If σ = {σi : i ∈ I} is a partition of the set P of all prime numbers, a subgroup H of a finite group G is said to be σ-subnormal in G if H can be joined to G by means of a chain of subgroups H = H0 ⊆ H1 ⊆ · · · ⊆ Hn = G such that either Hi−1 normal in Hi or Hi/ CoreHi (Hi−1) is a σ j-group for some j ∈ I, for every i = 1, . . . , n. If σ = {{2}, {3}, {5}, ...} is the minimal partition, then the σ-subnormality reduces to the classical subgroup embedding property of subnormality. A finite group X is said to be a Schmidt group if X is not nilpotent and every proper subgroup of X is nilpotent. Every non-nilpotent finite group G has Schmidt subgroups and a detailed knowledge of their embedding in G can provide a deep insight into its structure. In this paper, a complete description of a finite group with σ-subnormal Schmidt subgroups is given. It answers a question posed by Guo, Safonova and Skiba.