On σ-Permutably Embedded Subgroups of Finite Groups

Let sigma = {sigma(i): i I} be some partition of the set of all primes P, G be a finite group and sigma(G) = {sigma(i): sigma(i) (G)O}. A set H of subgroups of G is said to be a complete Hall sigma-set of G if every non-identity member of H is a Hall sigma(i)-subgroup of G and H contains exactly one Hall sigma(i)-subgroup of G for every sigma(i) sigma(G). G is said to be sigma-full if G possesses a complete Hall sigma-set. A subgroup H of G is sigma-permutable in G if G possesses a complete Hall sigma-set H such that HA(x)= A(x)H for all A H and all x G. A subgroup H of G is sigma-permutably embedded in G if H is sigma-full and for every sigma(i) sigma(H), every Hall sigma(i)-subgroup of H is also a Hall sigma(i)-subgroup of some sigma-permutable subgroup of G.By using the sigma-permutably embedded subgroups, we establish some new criteria for a group G to be soluble and supersoluble, and also give the conditions under which a normal subgroup of G is hypercyclically embedded. Some known results are generalized.