Geodesic
The influence of weakly-supplemented subgroups on the structure of finite groups
Czechoslovak Mathematical Journal, Tome 64 (2014) no. 1, pp. 173-182
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A subgroup $H$ of a finite group $G$ is weakly-supplemented in $G$ if there exists a proper subgroup $K$ of $G$ such that $G=HK$. In the paper it is proved that a finite group $G$ is $p$-nilpotent provided $p$ is the smallest prime number dividing the order of $G$ and every minimal subgroup of $P\cap G'$ is weakly-supplemented in $N_{G}(P),$ where $P$ is a Sylow $p$-subgroup of $G$. As applications, some interesting results with weakly-supplemented minimal subgroups of $P\cap G'$ are obtained.
A subgroup $H$ of a finite group $G$ is weakly-supplemented in $G$ if there exists a proper subgroup $K$ of $G$ such that $G=HK$. In the paper it is proved that a finite group $G$ is $p$-nilpotent provided $p$ is the smallest prime number dividing the order of $G$ and every minimal subgroup of $P\cap G'$ is weakly-supplemented in $N_{G}(P),$ where $P$ is a Sylow $p$-subgroup of $G$. As applications, some interesting results with weakly-supplemented minimal subgroups of $P\cap G'$ are obtained.
DOI : 10.1007/s10587-014-0092-y
Classification : 20D10, 20D20
Keywords: weakly-supplemented subgroup; $p$-nilpotent group; supersolvable group 
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Kong, Qingjun; Liu, Qingfeng. The influence of weakly-supplemented subgroups on the structure of finite groups. Czechoslovak Mathematical Journal, Tome 64 (2014) no. 1, pp. 173-182. doi: 10.1007/s10587-014-0092-y

[1] Arad, Z., Ward, M. B.: New criteria for the solvability of finite groups. J. Algebra 77 (1982), 234-246. | DOI | MR | Zbl

[2] Asaad, M., Ramadan, M., Shaalan, A.: Influence of $\pi$-quasinormality on maximal subgroups of Sylow subgroups of Fitting subgroup of a finite group. Arch. Math. 56 (1991), 521-527. | DOI | MR | Zbl

[3] Ballester-Bolinches, A., Guo, X.: On complemented subgroups of finite groups. Arch. Math. 72 (1999), 161-166. | DOI | MR | Zbl

[4] Doerk, K.: Minimal nicht überauflösbare, endliche Gruppen. German Math. Z. 91 (1966), 198-205. | DOI | MR | Zbl

[5] Hall, P.: A characteristic property of soluble groups. J. Lond. Math. Soc. 12 (1937), 188-200. | MR | Zbl

[6] Hall, P.: Complemented groups. J. Lond. Math. Soc. 12 (1937), 201-204. | DOI | MR | Zbl

[7] Huppert, B.: Endliche Gruppen I. German Die Grundlehren der Mathematischen Wissenschaften in Einzeldarstellungen 134 Springer, Berlin (1967). | DOI | MR | Zbl

[8] Kang, P.: Minimal subgroups and the structure of finite groups. Ric. Mat. 62 (2013), 91-95. | DOI | MR

[9] Li, D., Guo, X.: The influence of $c$-normality of subgroups on the structure of finite groups. II. Commun. Algebra 26 (1998), 1913-1922. | DOI | MR | Zbl

[10] Robinson, D. J. S.: A Course in the Theory of Groups. Graduate Texts in Mathematics 80 Springer, New York (1993). | MR

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