Czechoslovak Mathematical Journal, online first, 10 pp.

Finite groups whose all proper subgroups are $\mathcal{C}$-groups

Pengfei Guo, Jianjun Liu

Received October 16, 2016.   First published October 20, 2017.

Pengfei Guo, School of Mathematics and Statistics, Hainan Normal University, No. 99 Longkun South Road, Haikou 571158, Hainan, P. R. China, e-mail: guopf999@163.com; Jianjun Liu (corresponding author), School of Mathematics and Statistics, Southwest University, No. 2 Tiansheng Road, Beibei 400715, Chongqing, P. R. China, e-mail: liujj198123@163.com

Abstract: A group $G$ is said to be a $\mathcal{C}$-group if for every divisor $d$ of the order of $G$, there exists a subgroup $H$ of $G$ of order $d$ such that $H$ is normal or abnormal in $G$. We give a complete classification of those groups which are not $\mathcal{C}$-groups but all of whose proper subgroups are $\mathcal{C}$-groups.

Keywords: normal subgroup; abnormal subgroup; minimal non-$\mathcal{C}$-group

Classification (MSC 2010): 20D10, 20E34

DOI: 10.21136/CMJ.2017.0542-16

Full text available as PDF.


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