ON WEAKLY H-PERMUTABLE 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, G be a finite group and sigma(G) = {sigma(i)vertical bar sigma(i) boolean AND pi(G) (sic) (sic)}. G is said to be sigma-primary if vertical bar sigma(G)vertical bar <= 1. A subgroup H of G is said to be sigma-subnormal in G if there exists a subgroup chain H = H-0 <= H-1 <= center dot center dot center dot <= H-t = G such that either Hi-1 is normal in H-i or H-i/(Hi-1)H-i is sigma-primary for all i = 1; . . . t. 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 for some i and H contains exactly one Hall sigma(i)-subgroup of G for every sigma(i) is an element of sigma(G). Let H be a complete Hall sigma-set of G. A subgroup H of G is said to be H-permutable if HA = AH for all A is an element of H. We say that a subgroup H of G is weakly H-permutable in G if there exists a sigma-subnormal subgroup T of G such that G = HT and H boolean AND T <= H-H, where H-H is the subgroup of H generated by all those subgroups of H which are H-permutable. By using the weakly H-permutable subgroups, we establish some new criteria for a group G to be sigma-soluble and supersoluble, and we also give the conditions under which a normal subgroup of G is hypercyclically embedded.