ON m-σ-EMBEDDED SUBGROUPS OF FINITE GROUPS

Let sigma = {sigma(i)vertical bar i is an element of I} be some partition of the set of all primes P and G be a finite group. A group is said to be sigma-primary if it is a finite sigma(i)-group for some i. A subgroup A of G is said to be sigma-subnormal in G if there is a subgroup chain A = A(0) <= A(1) <= . . . = A(t) = G such that either A(i-1) (sic) A(i) or A(i)/(A(i-1)) A(i) is sigma-primary for all i = 1, . . . , t. A subgroup S of G is m-sigma-permutable in G if S = < M, B > for some modular subgroup M and sigma-permutable subgroup B of G. We say that a subgroup H of G is m-sigma-embedded in G if there exist an m-sigma-permutable subgroup S and a sigma-subnormal subgroup T of G such that H-G = HT and H boolean AND T <= S <= H, where H-G = < H-x vertical bar x is an element of G > is the normal closure of H in G. In this paper, we study the properties of m-sigma-embedded subgroups and use them to determine the structure of finite groups. Some known results are generalized.