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Keywords:
weakly-supplemented subgroup; solvable group; finite group
weakly-supplemented subgroup; solvable group; finite group
Summary:
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.
References:
[1] Hall, P.: 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] Huppert, B.: Endliche Gruppen. I. Die Grundlehren der mathematischen Wissenschaften in Einzeldarstellungen 134. Springer, Berlin (1967), German. DOI 10.1007/978-3-642-64981-3 | MR 0224703 | Zbl 0217.07201
[3] Zhou, Q.: 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
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