Some Sufficient Conditions of Soluble Groups

A subgroup H of a group G is said to be weakly cnormal in G if there exists a subnormal subgroup K of G such that G=HK and H∩K≤HG, where HG=CoreG∩x∈GHx is the largest normal subgroup of G that contained in H. In this paper, the authors give some sufficient conditions under which a group G is soluble one by using weakly cnormality of πHall subgroups, Sylow subgroup with odd order and second maximal subgroups of G.