Czechoslovak Mathematical Journal, Vol. 72, No. 4, pp. 1045-1046, 2022


A note on weakly-supplemented subgroups and the solvability of finite groups

Xin Liang, Baiyan Xu

Received September 6, 2021.   Published online March 18, 2022.

Abstract:  Suppose that $G$ is a finite group and $H$ is a subgroup of $G$. The subgroup $H$ is said to be weakly-supplemented in $G$ if there exists a proper subgroup $K$ of $G$ such that $G=HK$. In this note, by using the weakly-supplemented subgroups, we point out several mistakes in the proof of Theorem 1.2 of Q. Zhou (2019) and give a counterexample.
Keywords:  weakly-supplemented subgroup; solvable group; finite group
Classification MSC:  20D10, 20D20
DOI:  10.21136/CMJ.2022.0326-21


References:
[1] P. Hall: A characteristic property of soluble groups. J. Lond. Math. Soc. 12 (1937), 198-200. DOI 10.1112/jlms/s1-12.2.198 | MR 1575073 | Zbl 0016.39204
[2] B. Huppert: Endliche Gruppen. I. Die Grundlehren der mathematischen Wissenschaften in Einzeldarstellungen 134. Springer, Berlin (1967). (In German.) DOI 10.1007/978-3-642-64981-3 | MR 0224703 | Zbl 0217.07201
[3] Q. Zhou: On weakly-supplemented subgroups and the solvability of finite groups. Czech. Math. J. 69 (2019), 331-335. DOI 10.21136/CMJ.2018.0301-17 | MR 3959947 | Zbl 07088787

Affiliations:   Xin Liang (corresponding author), Baiyan Xu Basic Research Section, Changchun Guanghua University, Road Economic & Technological Development Zone, Changchun, Jilin 130033, P. R. China, e-mail: mathxj2012@163.com


 
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