Article
Full entry |
PDF
(0.2 MB)
Feedback
PDF
(0.2 MB)
Feedback
Keywords:
normal subgroup; abnormal subgroup; minimal non-$\mathcal {C}$-group
normal subgroup; abnormal subgroup; minimal non-$\mathcal {C}$-group
Summary:
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.
References:
[1] Ballester-Bolinches, A., Esteban-Romero, R.: On minimal non-supersoluble groups. Rev. Mat. Iberoam. 23 (2007), 127-142. DOI 10.4171/RMI/488 | MR 2351128 | Zbl 1126.20013
[2] Ballester-Bolinches, A., Esteban-Romero, R., Robinson, D. J. S.: 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] Doerk, K.: Minimal nicht überauflösbare, endliche Gruppen. Math. Z. 91 (1966), 198-205 German. DOI 10.1007/BF01312426 | MR 0191962 | Zbl 0135.05401
[4] Doerk, K., Hawkes, T.: Finite Soluble Groups. De Gruyter Expositions in Mathematics 4, Walter de Gruyter, Berlin (1992). DOI 10.1515/9783110870138 | MR 1169099 | Zbl 0753.20001
[5] Laffey, T. J.: 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] Liu, J., Li, S., He, J.: 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] Miller, G. A., Moreno, H. C.: Non-abelian groups in which every subgroup is abelian. Trans. Amer. Math. Soc. 4 (1903), 398-404 \99999JFM99999 34.0173.01. DOI 10.1090/S0002-9947-1903-1500650-9 | MR 1500650
[8] Robinson, D. J. S.: 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] Šmidt, O. J.: Über Gruppen, deren sämtliche Teiler spezielle Gruppen sind. Math. Sbornik 31 (1924), 366-372 Russian with German résumé \99999JFM99999 50.0076.04.
Search
Partner of
