Czechoslovak Mathematical Journal

Czechoslovak Mathematical Journal, Vol. 68, No. 2, pp. 513-522, 2018

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

Pengfei Guo, Jianjun Liu

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

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: 20D10, 20E34

DOI: 10.21136/CMJ.2017.0542-16

PDF available at: Springer Institute of Mathematics CAS Digital Mathematics Library

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Affiliations: 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

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