Abstract
A p-group G is called a \(\mathcal {NCC}\)-p-group if \(N_G(\langle x\rangle )/\langle x\rangle \) is cyclic for every non-normal cyclic subgroup \(\langle x\rangle \) in G. In this paper, we give a complete classification of finite \(\mathcal {NCC}\)-p-groups.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
Berkovich, Y.: Groups of Prime Power Order, vol. 1. Walter de Gruyter, Berlin (2008)
Blackburn, N.: Generalizations of certain elementary theorems on \(p\)-groups. Proc. London Math. Soc. 11(3), 1–22 (1961)
Cao, J.J., Guo, X.Y.: Finite soluble groups in which the normalizer of every non-normal cyclic subgroup is maximal. J. Group Theory 17, 671–687 (2014)
Cutolo, G., Smith, H., Wiegold, J.: The nilpotency class of \(p\)-groups in which subgroups have few conjugates. J. Algebra 300, 160–170 (2006)
Huppert, B.: Endliche Gruppen I. Springer, Berlin (1967)
Li, X.H., Zhang, J.Q.: Finite \(p\)-groups with non-normal subgroups of index \(p\) in their normalizers. Comm. Algebra 39(6), 2037–2043 (2011)
Newman, M.F., Xu, M.: Metacyclic groups of prime-power order (Research announcement). Adv. Math. 17, 106–107 (1988)
Xu, M., An, L., Zhang, Q.: Finite \(p\)-groups all of whose non-abelian proper subgroups are generated by two elements. J. Algebra 319, 3603–3620 (2008)
Zhang, J.Q.: Finite \(p\)-groups all of whose nonnormal subgroups posses special normalizers. Bull. Malaysian Math. Sci. Soc. 40, 279–293 (2017)
Zhang, Q.H., Cao, J.: Normalizers of nonnormal subgroups of finite p-groups. J. Korean Math. Soc. 49(1), 201–221 (2012)
Zhang, X., Guo, X.: Finite \(p\)-groups whose non-normal cyclic subgroups have small index in their normalizers. J. Group theory 15, 641–659 (2012)
Acknowledgements
The research of the work is supported by “The Science & Technology Development Fund of Tianjin Education Commission for Higher Education (2019KJ141)” and “Tianjin Technical Expert Project under grant (21YDTPJC00120)”. The author would like to thank the referee for his or her valuable suggestions and useful comments which contributed to the final version of this paper.
Author information
Authors and Affiliations
Corresponding author
Ethics declarations
Conflict of interest
The author declares no conflict of interest.
Additional information
Publisher's Note
Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.
The research of the work was supported by “The Science & Technology Development Fund of Tianjin Education Commission for Higher Education(2019KJ141)” and “Tianjin Technical Expert Project under grant(21YDTPJC00120)”.
Rights and permissions
Springer Nature or its licensor (e.g. a society or other partner) holds exclusive rights to this article under a publishing agreement with the author(s) or other rightsholder(s); author self-archiving of the accepted manuscript version of this article is solely governed by the terms of such publishing agreement and applicable law.
About this article
Cite this article
Wang, J. Finite p-groups whose non-normal cyclic subgroups have cyclic quotient groups in their normalizers. Ricerche mat (2023). https://doi.org/10.1007/s11587-023-00797-7
Received:
Accepted:
Published:
DOI: https://doi.org/10.1007/s11587-023-00797-7