A subgroup H of a group G is said to be weakly s-supplemented in G if H has a supplement T in G such that H ∩ T HsG, where HsG is the largest s-permutable subgroup of G contained in H. This paper constructs an exampl...A subgroup H of a group G is said to be weakly s-supplemented in G if H has a supplement T in G such that H ∩ T HsG, where 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.展开更多
AbstractLet G be a finite group and H a subgroup of G. Then H is said to be S-permutable in G if HP = PH for all Sylow subgroups P of G. Let H sG be the subgroup of H generated by all those subgroups of H which are S-...AbstractLet G be a finite group and H a subgroup of G. Then H is said to be S-permutable in G if HP = PH for all Sylow subgroups P of G. Let H sG be the subgroup of H generated by all those subgroups of H which are S-permutable in G. Then we say that H is S-embedded in G if G has a normal subgroup T and an S-permutable subgroup C such that T ∩ H ? H sG and HT = C.Our main result is the followingTheorem AA group G is supersoluble if and only if for every non-cyclic Sylow subgroup P of the generalized Fitting subgroup F*(G) of G, at least one of the following holds: Every maximal subgroup of P is S-embedded in G.Every cyclic subgroup H of P with prime order or order 4 (if P is a non-abelian 2-group and H ? Z ∞ (G)) is S-embedded in G.展开更多
Let A be a subgroup of a group G and X be a nonempty subset of G. A is said to be X-semipermutable in G if A has a supplement T in G such that A is X-permutable with every subgroup of T. In this paper, we investigate ...Let A be a subgroup of a group G and X be a nonempty subset of G. A is said to be X-semipermutable in G if A has a supplement T in G such that A is X-permutable with every subgroup of T. In this paper, we investigate further the influence of X-semipermutability of some subgroups on the structure of finite groups. Some new criteria for a group G to be supersoluble or p-nilpotent are obtained.展开更多
基金supported by National Natural Science Foundation of China (Grant No. 10771180)
文摘A subgroup H of a group G is said to be weakly s-supplemented in G if H has a supplement T in G such that H ∩ T HsG, where 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.
基金supported by National Natural Science Foundation of China (Grant No. 10771180)
文摘AbstractLet G be a finite group and H a subgroup of G. Then H is said to be S-permutable in G if HP = PH for all Sylow subgroups P of G. Let H sG be the subgroup of H generated by all those subgroups of H which are S-permutable in G. Then we say that H is S-embedded in G if G has a normal subgroup T and an S-permutable subgroup C such that T ∩ H ? H sG and HT = C.Our main result is the followingTheorem AA group G is supersoluble if and only if for every non-cyclic Sylow subgroup P of the generalized Fitting subgroup F*(G) of G, at least one of the following holds: Every maximal subgroup of P is S-embedded in G.Every cyclic subgroup H of P with prime order or order 4 (if P is a non-abelian 2-group and H ? Z ∞ (G)) is S-embedded in G.
基金supported by National Natural Science Foundation of China (Grant Nos. 10771172, 10771180)
文摘Let A be a subgroup of a group G and X be a nonempty subset of G. A is said to be X-semipermutable in G if A has a supplement T in G such that A is X-permutable with every subgroup of T. In this paper, we investigate further the influence of X-semipermutability of some subgroups on the structure of finite groups. Some new criteria for a group G to be supersoluble or p-nilpotent are obtained.
