H is called an Mp-embedded subgroup of G, if there exists a pnilpotent subgroup B of G such that Hp E Sylp（B） and B is Mp-supplemented in G. In this paper, by considering prime divisor 3, 5, or 7, we use Mp-embedded...H is called an Mp-embedded subgroup of G, if there exists a pnilpotent subgroup B of G such that Hp E Sylp（B） and B is Mp-supplemented in G. In this paper, by considering prime divisor 3, 5, or 7, we use Mp-embedded property of primary subgroups to investigate the solvability of finite groups. The main result is follows. Let E be a normal subgroup of G, and let P be a Sylow 5-subgroup of E. Suppose that 1 〈 d ≤ ｜P｜ and d divides ｜P｜. If every subgroup H of P with ｜H｜ = d is M5-embedded in G, then every composition factor of E satisfies one of the following conditions： （1） I/C is cyclic of order 5, （2） I/C is 5＇-group, （3） I/C A5.展开更多
基金This work was supported in part by the National Natural Science Foundation of China （Grant Nos. 11271016, 11501235）, the Qing Lan Project of Jiangsu Province, the High-level Personnel of Support Program of Yangzhou University, the Natural Science Foundation of Xinjiang Uygur Autonomous Region （Grant No. 2016D01C387）, and the Key Natural Science Foundation of Anhui Education Commission （KJ2017A569）.
文摘H is called an Mp-embedded subgroup of G, if there exists a pnilpotent subgroup B of G such that Hp E Sylp（B） and B is Mp-supplemented in G. In this paper, by considering prime divisor 3, 5, or 7, we use Mp-embedded property of primary subgroups to investigate the solvability of finite groups. The main result is follows. Let E be a normal subgroup of G, and let P be a Sylow 5-subgroup of E. Suppose that 1 〈 d ≤ ｜P｜ and d divides ｜P｜. If every subgroup H of P with ｜H｜ = d is M5-embedded in G, then every composition factor of E satisfies one of the following conditions： （1） I/C is cyclic of order 5, （2） I/C is 5＇-group, （3） I/C A5.
