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.
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References
P. Hall: A characteristic property of soluble groups. J. Lond. Math. Soc. 12 (1937), 198–200.
B. Huppert: Endliche Gruppen. I. Die Grundlehren der mathematischen Wissenschaften in Einzeldarstellungen 134. Springer, Berlin, 1967. (In German.)
Q. Zhou: On weakly-supplemented subgroups and the solvability of finite groups. Czech. Math. J. 69 (2019), 331–335.
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The authors are very grateful to the referee who read the manuscript carefully and provided a lot of valuable suggestions and useful comments. It should be said that we would not have been able to polish the final version of this paper well without his or her outstanding efforts.
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Liang, X., Xu, B. A note on weakly-supplemented subgroups and the solvability of finite groups. Czech Math J 72, 1045–1046 (2022). https://doi.org/10.21136/CMJ.2022.0326-21
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DOI: https://doi.org/10.21136/CMJ.2022.0326-21