On inequalities of subgroups and the structure of finite groups

Abstract

Let G be a group and H be a subgroup of G. We say that H is weakly -supplemented in G if G has a subgroup T such that HT = G and H ∩ T ≤ (H), where (H) denotes the Frattini subgroup of H. In this paper, properties of this new kind of inequalities of subgroups are investigated and new characterizations of nilpotency and supersolubility of finite groups in terms of the new inequalities are obtained. MSC: 20D10; 20D15; 20D20

On nearly SΦ-normal subgroups of finite groups

“…Recall that a subgroup H of a group G is said to be s-permutable (or s-quasinormal ) [17] in G if HP = P H for all Sylow subgroups P of G. A subgroup H of G is said to be c-normal in G [30] if there exists a normal subgroup T of G such that HT = G and H ∩ T ≤ H G , where H G is the largest normal subgroup of G contained in H. A subgroup H of G is said to be Φ-S-supplemented in G [18,19] if there exists a subnormal subgroup T of G such that HT = G and H ∩ T ≤ Φ(H), where Φ(H) is Frattini subgroup of H. A subgroup H of G is said to be weakly Φ-supplemented in G [20] if there exists a subgroup T of G such that HT = G and H ∩ T ≤ Φ(H). A subgroup H of G is said to be nearly s-normal [32] in G if there exists a normal subgroup T of G such that HT ¢ G and H ∩ T ≤ H sG , where H sG is the subgroup of H generated by all those subgroups of H which are s-permutable in G. A subgroup H of G is said to be weakly SΦ-supplemented [33] in G if there exists a subgroup T of G such that G = HT and H ∩ T ≤ Φ(H)H sG .…”