Solvability of finite groups

doi:10.1007/s11464-017-0643-2

Front. Math. China

2017, Vol. 12

Issue (6) : 1501-1514 https://doi.org/10.1007/s11464-017-0643-2

RESEARCH ARTICLE

Solvability of finite groups

Jia ZHANG¹, Baijun GAO², Long MIAO³(

)

¹. School of Mathematics and Information, China West Normal University, Nanchong 637009, China
². School of Mathematics and Statistics, Yili Normal University, Yining 835000, China
³. School of Mathematical Sciences, Yangzhou University, Yangzhou 225002, China

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Abstract

H is called an $M p$ -embedded subgroup of G, if there exists a pnilpotent subgroup B of G such that H_p ∈ Syl_p (B) and B is $M p$ -supplemented in G. In this paper, by considering prime divisor 3, 5, or 7, we use $M p$ -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 $M 5$ -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) $I / C ≅ A 5$

Keywords Composition factor Mp-embedded subgroup')" href="#">

M p

-embedded subgroup primary subgroup solvable

Corresponding Author(s): Long MIAO

Issue Date: 27 November 2017

Cite this article:

Jia ZHANG,Baijun GAO,Long MIAO. Solvability of finite groups[J]. Front. Math. China, 2017, 12(6): 1501-1514.

URL:

https://academic.hep.com.cn/fmc/EN/10.1007/s11464-017-0643-2
https://academic.hep.com.cn/fmc/EN/Y2017/V12/I6/1501

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