On the Supersolvablity of Finite Groups
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On the Supersolvablity of Finite Groups Xianhe Zhao1,2
· Junzhen Sui1 · Ruifang Chen1 · Qin Huang2
Received: 16 September 2019 / Revised: 11 November 2019 / Accepted: 30 November 2019 © Iranian Mathematical Society 2020
Abstract In this paper, we investigate C SS-quasinormality of maximal subgroups of Sylow subgroups of some Fitting subgroup and obtain a sufficient condition about the supersolvablity of finite groups, such that some known results about c-normality and SS-quasinormality are generalized and unified. Keywords C SS-subgroups · c-normal subgroups · SS-quasinormal subgroups · Supersolvable groups · Saturated formations Mathematics Subject Classification 20D10 · 20D20
Communicated by Mohammad Reza Darafsheh. The research has been supported by the National Natural Science Foundation of China (nos. 11501176, 11901169), the Project for University Young Key Teachers from Henan Province Education Department, the project for high quality courses of postgraduate education in Henan Province and Key Laboratory of Applied Mathematics of Fujian Province University (Putian University, no. SX201902).
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Xianhe Zhao [email protected] Junzhen Sui [email protected] Ruifang Chen [email protected] Qin Huang [email protected]
1
School of Mathematics and Information Sciences, Engineering Laboratory for Big Data Statistical Analysis and Optimal Control, Henan Normal University, Xinxiang 453007, Henan, People’s Republic of China
2
Key Laboratory of Applied Mathematics (Putian University), Fujian Province University, Putian 351100, Fujian, People’s Republic of China
123
Bulletin of the Iranian Mathematical Society
1 Introduction All groups considered in this paper are finite. Let G be a group, we employ the notation F(G) to denote the Fitting subgroup of G, and U to denote the supersolvable group formation. It is well known to all that the supersolvability of a group G has been an important topic in finite group theory, and many authors have studied this problem from different perspectives [1,3,5,7,9,10] and so on. Among these results, [5,7] investigate the supersolvability of G using c-normality and SS-quasinormality, separately. Recall a subgroup H of a group G is said to be c-normal in G if there exists a normal subgroup N of G, such that G = H N with H ∩ N ≤ HG ≤ Cor eG (H ) (see [12]); a subgroup H of G is said to be an SS-quasinormal subgroup of G if there is a supplement B of H to G, such that H permutes with every Sylow subgroup of B (see [8]). It is worth mentioning that, using the concept of c-normality, the authors prove that: Theorem A [5, Theorem 2] A solvable group G is supersolvable, if there exists such a normal subgroup H of G that G/H is supersolvable and all maximal subgroups of any Sylow subgroup of F(H ) are c-normal in G. In virtue of SS-quasinormality, the authors admit that: Theorem B [7, Theorem 3.1] G is supersolvable, if G has such a solvable normal subgroup H that G/H is supersolvable and all maximal subgroups of any Sylow subgroup of F(H ) are SS-quasinormal in G. Notice that, re
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