Abstract. Let H be a subgroup of a group G. Then, we call H weakly s-supplemented in G if G has a subgroup T such that HT = G and H ∩ T ≤ H sG , where H sG is the largest s-permutable subgroup of G contained in H. In this paper, we use the weakly s-supplemented subgroups to characterize the structure of groups. A series of known results in the literature are unified and generalized.2010 Mathematics Subject Classification. 20D10, 20D25.
A subgroup H of a group G is said to be weakly s-supplemented in G if H has a supplementwhere HsG is the largest s-permutable subgroup of G contained in H. This paper constructs an example to show that the open questions 6.3 and 6.4 in J Algebra, 315: 192-209 (2007) have negative solutions, and shows that in many cases Question 6.4 is positive. A series of known results are unified and generalized.
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
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