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

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.

[1] A. Ballester-Bolinches, R. Esteban-Romero: On minimal non-supersoluble groups. Rev. Mat. Iberoam. 23 (2007), 127-142. DOI 10.4171/RMI/488 | MR 2351128 | Zbl 1126.20013
[2] A. Ballester-Bolinches, R. Esteban-Romero, D. J. S. Robinson: On finite minimal non-nilpotent groups. Proc. Am. Math. Soc. 133 (2005), 3455-3462. DOI 10.1090/S0002-9939-05-07996-7 | MR 2163579 | Zbl 1082.20006
[3] K. Doerk: Minimal nicht überauflösbare, endliche Gruppen. Math. Z. 91 (1966), 198-205. (In German.) DOI 10.1007/BF01312426 | MR 0191962 | Zbl 0135.05401
[4] K. Doerk, T. Hawkes: Finite Soluble Groups. De Gruyter Expositions in Mathematics 4, Walter de Gruyter, Berlin (1992). DOI 10.1515/9783110870138 | MR 1169099 | Zbl 0753.20001
[5] T. J. Laffey: A lemma on finite $p$-groups and some consequences. Proc. Camb. Philos. Soc. 75 (1974), 133-137. DOI 10.1017/S0305004100048350 | MR 0332961 | Zbl 0277.20022
[6] J. Liu, S. Li, J. He: CLT-groups with normal or abnormal subgroups. J. Algebra 362 (2012), 99-106. DOI 10.1016/j.jalgebra.2012.03.042 | MR 2921632 | Zbl 1261.20027
[7] G. A. Miller, H. C. Moreno: Non-abelian groups in which every subgroup is abelian. Trans. Amer. Math. Soc. 4 (1903), 398-404. DOI 10.1090/S0002-9947-1903-1500650-9 | MR 1500650 | JFM 34.0173.01
[8] D. J. S. Robinson: A Course in the Theory of Groups. Graduate Texts in Mathematics 80, Springer, New York (1982). DOI 10.1007/978-1-4684-0128-8 | MR 0648604 | Zbl 0483.20001
[9] O. J. Šmidt: Über Gruppen, deren sämtliche Teiler spezielle Gruppen sind. Math. Sbornik 31 (1924), 366-372. (In Russian with German résumé.) JFM 50.0076.04

Access to the full text of journal articles on this site is restricted to the subscribers of Myris Trade. To activate your access, please contact Myris Trade at
Subscribers of Springer need to access the articles on their site, which is

[List of online first articles] [Contents of Czechoslovak Mathematical Journal] [Full text of the older issues of Czechoslovak Mathematical Journal at DML-CZ]