Czechoslovak Mathematical Journal, first online, pp. 1-10, 2017

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

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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:; Jianjun Liu (corresponding author), School of Mathematics and Statistics, Southwest University, No. 2 Tiansheng Road, Beibei 400715, Chongqing, P. R. China, e-mail:

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