A note on weakly -n-embedded subgroups of -full groups

Let sigma = {sigma(i) : i is an element of I} be a partition of the set P of all prime numbers. A subgroup H of a finite group G is called s-subnormal in G if there is a chain of subgroups H = H-0 subset of H-1 subset of center dot center dot center dot subset of H-n = G where, for every i = 1, ..., n, Hi-1 normal in H-i or H-i/(Hi-1)H-i is a sigma(i)-group for some i is an element of I. A subgroup H of G is said to be weakly sigma-n-embedded in G, if there is a sigma-subnormal subgroup T of G such that H-G = HT and H boolean AND T <= H-sigma G, where the subgroup H-sigma G is generated by all s-subnormal subgroups of G contained in H. In this paper, we study the properties of weakly sigma-n-embedded subgroups and use them to determine the structure of finite groups. Some known results are gerneralized.