Finite Groups with sigma -Subnormal Schmidt Subgroups

Abstract

If sigma = { {sigma }_{i} : i in I } is a partition of the set mathbb {P} of all prime numbers, a subgroup H of a finite group G is said to be sigma -subnormal in G if H can be joined to G by means of a chain of subgroups H=H_{0} subseteq H_{1} subseteq cdots subseteq H_{n}=G such that either H_{i-1} normal in H_{i} or H_{i}/{{,mathrm{Core},}}_{H_{i}}(H_{i-1}) is a {sigma }_{j}-group for some j in I, for every i=1, ldots , n. If sigma = {{2}, {3}, {5}, ... } is the minimal partition, then the sigma -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 sigma -subnormal Schmidt subgroups is given. It answers a question posed by Guo, Safonova and Skiba.