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

PDF available at:  Springer   Institute of Mathematics CAS

[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

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:

PDF available at: