Abstract
Let G be a finite p-group. Assume that \(\nu (G)\) and \(\nu _c(G)\) denote the number of conjugacy classes of non-normal subgroups and non-normal cyclic subgroups of G, respectively. In this paper, we completely classify the finite p-groups with \(\nu _c=p\) or \(p+1\) for an odd prime number p. Also, we classify the groups G with \(\nu (G)=\nu _c(G)=p^i, i\ge 1\).
This is a preview of subscription content, access via your institution.
We’re sorry, something doesn't seem to be working properly.
Please try refreshing the page. If that doesn't work, please contact support so we can address the problem.
References
- 1.
Berkovich, Y.: Groups of Prime Power Order I. Springer, Berlin (2008)
- 2.
Blackburn, N.: Finite groups in which the normal subgroups have non-trivial intersection. J. Algebra 3, 30–37 (1966)
- 3.
Brandl, R.: Groups with few non-normal subgroups. Commun. Algebra 6, 2091–2098 (1995)
- 4.
Brandl, R.: Conjugacy classes of non-normal subgroups of finite p-groups. Isr. J. Math. 195, 473–479 (2013)
- 5.
Bozikov, Z., Janko, Z.: A complete classification of finite p-groups all of whose non-cyclic subgroups are normal. Glasnik Mat. 44(1), 177–185 (2009)
- 6.
Cohn, J.H.E.: On n-sum groups. Math. Scand. 75, 44–58 (1994)
- 7.
Fernandez-Alcober, G.A., Legarreta, L.: The finite \(p\)-groups with \(p\) conjugacy classes of non-normal subgroups. Isr. J. Math. 180, 189–192 (2010)
- 8.
Huppert, B.: Endliche Gruppen I. Springer, Berlin (1967)
- 9.
Li, L., Qu, H.: The number of conjugacy classes of non-normal subgroups of finite p-groups. J. Algebra 1 (2016)
- 10.
Mousavi, H.: On finite groups with few non-normal subgroups. Commun. Algebra 7, 3143–3151 (1999)
- 11.
Mousavi, H.: Nilpotent groups with three conjugacy classes of non-normal subgroups. Bull. Iran. Math. Soc. 5, 1291–1300 (2014)
- 12.
Mousavi, H.: Non-nilpotent groups with three conjugacy classes of non-normal subgroups. Int. J. Group Theory 3(2), 1–7 (2014)
- 13.
Mousavi, H., Ahmadi, H.: A bound on the number of conjugacy classes of non-normal cyclic subgroups of a finite \(p\)-group (submitted)
- 14.
Oggionni, D., Ponzoni, G., Zambelli, V.: Groups with few non-normal cyclic subgroups. Note Mat. 30(2), 121–133 (2010)
- 15.
Passman, D.S.: Non-normal subgroups of \(p\)-groups. J. Algebra 15(3), 352–370 (1970)
- 16.
Poland, J., Rhemtulla, A.: The number of conjugacy classes of non-normal subgroups in nilpotent groups. Commun. Algebra 24(10), 3237–3245 (1996)
- 17.
Redei, L.: Das Schiefe Produkt in der Gruppentheorie. Comment. Math. Helv. 20, 225–264 (1947)
- 18.
Shirong, L.: The number of conjugacy classes of non-normal cyclic subgroups in nilpotent groups of odd order. J. Group Theory 1, 165–171 (1998)
Author information
Affiliations
Corresponding author
Additional information
Publisher's Note
Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.
Communicated by Mohammad Zarrin.
Supplementary Information
Below is the link to the electronic supplementary material.
Rights and permissions
About this article
Cite this article
Mousavi, H. On the Conjugacy Classes of Cyclic Non-normal Subgroups. Bull. Iran. Math. Soc. (2021). https://doi.org/10.1007/s41980-020-00502-6
Received:
Accepted:
Published:
Keywords
- Non-normal subgroups
- Conjugacy class of non-normal subgroups
Mathematics Subject Classification
- 20E45
- 20D25