Finite Groups with σ-Subnormal Schmidt Subgroups

If sigma = (sigma(i) : i is an element of I) is a partition of the set 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 subset of H-1 subset of . . . subset of H-n = G such that either Hi-1 normal in H-i or H-i/ Core(Hi) (Hi-1) is a sigma(j)-group for some j is an element of I, for every i = 1, . , 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.