Finite $p$-groups in which the normalizer of each nonnormal subgroup is small

Abstract

Let $G$ be a finite non-Dedekindian $p$-group which satisfies $N_G(H)=HZ(G)$ for each nonnormal subgroup $H$, and we call it an $NS$-group. In this paper, it is proved that an $NS$-group is the product of a minimal nonabelian group and the center.

Keywords

• non-Dedekindian $p$-group
• the center
• nonnormal subgroup

Project Number

The National Science Foundation of China (No. 12101135, 12071092,12371021, 12171302)

References

1. [1] Y. Berkovich, Groups of Prime Power Order, volume 1, Walter de Gruyter, Berlin, 2008.
2. [2] Y. Berkovich and Z. Janko, Groups of Prime Power Order, volume 2, Walter de Gruyter, Berlin, 2008.
3. [3] G. Cutolo, E. I. Khukhro, J. C. Lennox, J. Wiegold, S. Rinauro and H. Smith, Finite core-p p-groups, J. Algebra, 188, 701-719, 1997.
4. [4] G. Cutolo, H. Smith and J. Wiegold, Finite core-2 2-groups, J. Algebra, 237, 813-841, 2001.
5. [5] X. Li and J. Zhang, Finite p-groups with nonnormal subgroups of index p in their normalizers, Comm. Algebra, 39, 2037-2043, 2011.
6. [6] D. Yang, L. An and H. Lv, Finite p-groups all of whose nonnormal subgroups have bounded normal cores, Bull. Aust. Math. Soc., 101, 255-265, 2020.
7. [7] Q. Zhang and J. Gao, Nomalizers of nonnormal subgroups of finite p-groups, J. Korean Math. Soc., 49(1), 201-221, 2012.
8. [8] X. Zhang and X. Guo, Finite p-groups whose nonnormal cyclic subgroups have small index in their normalizers, J. Group theory, 15, 641-659, 2012.

9. [9] L. Zhao, L. Gong and X. Guo, Finite p-groups with a trivial core or the normal closure index p for every nonnormal abelian subgroup, Chinese Ann. Math. A, 42(4), 419-426, 2021.
10. [10] L. Zhao, Y. Li and L. Gong, Finite groups in which the cores of every nonnormal subgroups are trivial, Publ. Math. Debrecen, 93(3-4),511-516, 2018.
11. [11] L. Zhao, Y. Li, L. Gong and X. Guo, Finite p-groups in which the intersection of nonnormal subgroup and center has bounded order, Comm. Algebra, 51(9), 3825- 3833, 2023.
12. [12] L. Zhao, Y. Li, L. Gong and X. Guo, Finite p-groups in which the cores of all the nonnormal subgroups are in the center, J. Algebra. App. accepted.