Characterization of finite groups with a unique non-nilpotent proper subgroup

Document Type : Research Paper


Department of Mathematical Sciences, Isfahan University of Technology, P.O.Box 84156-83111, Isfahan, Iran


‎We characterize finite non-nilpotent groups $G$ with a unique non-nilpotent proper subgroup‎. ‎We show that $|G|$ has at most three prime divisors‎. ‎When $G$ is supersolvable we find the presentation of $G$ and when $G$ is non-supersolvable we show that either $G$ is a direct product of an Schmidt group and a cyclic group or a semi direct product of a $p$-group by a cyclic group of prime power order‎.


Main Subjects

[1] A. Ballester-Bolinches, R. Esteban-Romero, On minimal non-supersoluble group, Rev. Mat. Iberoamericana., 23 no.
1 (2007) 127–142.
[2] N. Itˆo, Note on (LM)-groups of finite order, Kdai Math. Sem. Rep., 3 (1951) 1–6.
[3] H. Kurzweil and B. Stellmacher, The theory of finite groups, An introduction, Translated from the 1998 German
original, Universitext, Springer-Verlag, New York, 2004.
[4] G. A. Miller and H. C. Moreno, Non-abelian groups in which every subgroup is abelian, Am. Math. Soc., 4 (1903)
[5] L. R´edei, Das Schiefe Produkt in der Gruppentheorie, Comment. Math. Helv., 20 (1947) 225–264.
[6] L. R´edei, Die endlichen einstufig nichtnilpotenten Gruppen, Publ. Math. Debrecen., 4 (1956) 303–324.
[7] J. S Rose, The influence on a finite group of its proper abnormal structure, J. London Math. Soc., 40 (1965) 348–361.
[8] D. J. S. Robinson, A course in the theory of groups, 2nd ed., Springer-Verlag, New York, 1996.
[9] F. G. Russo, Minimal non-nilpotent groups which are supersolvable, JP. J. Algebra Number Theory Appl., 18 (2010)
[10] O. Yu. Schmidt, Groups whose all subgroups are special, Math. Sb., 31 (1924) 366–372.
[11] B. Taeri and F. Tayanloo-Beyg, Finite groups with a unique non-abelian proper subgroup, J. Algebra Appl., (2019)
pp. 13.
[12] M. Zarrin, A generalization of Schmidt’s Theorem on groups with all subgroups nilpotent, Arch. Math., 99 (2012)