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FINITE p-GROUPS WHOSE NORMAL CLOSURES OF NON-NORMAL SUBGROUPS HAVE TWO ORDERS

  • Wang, Lifang (Department of Mathematics Shanxi Normal University)
  • Received : 2021.11.25
  • Accepted : 2022.05.13
  • Published : 2022.07.01

Abstract

We describe the structure of finite p-groups in which all normal closures of non-normal subgroups have two orders for p > 2.

Keywords

Acknowledgement

The author would like to express her sincere gratitude to the referee. Thanks to the referee for his or her valuable comments. Thanks Professor Qinhai Zhang for his frequent encouragement.

References

  1. L. An, Finite p-groups whose non-normal subgroups have few orders, Front. Math. China 13 (2018), no. 4, 763-777. https://doi.org/10.1007/s11464-018-0693-0
  2. Y. Berkovich, Groups of prime power order. Vol. 1, De Gruyter Expositions in Mathematics, 46, Walter de Gruyter GmbH & Co. KG, Berlin, 2008.
  3. L. L. Li, H. P. Qu, and G. Y. Chen, Central extension of inner abelian p-groups. I, Acta Math. Sinica (Chinese Ser.) 53 (2010), no. 4, 675-684.
  4. D. S. Passman, Nonnormal subgroups of p-groups, J. Algebra 15 (1970), 352-370. https://doi.org/10.1016/0021-8693(70)90064-5
  5. L. Wang and H. Qu, Finite groups in which the normal closures of non-normal subgroups have the same order, J. Algebra Appl. 15 (2016), no. 7, 1650125, 15 pp. https://doi.org/10.1142/S0219498816501255
  6. M. Y. Xu, A theorem on metabelian p-groups and some consequences, Chinese Ann. Math. Ser. B 5 (1984), no. 1, 1-6.
  7. Q. Zhang, X. Guo, H. Qu, and M. Xu, Finite groups which have many normal subgroups, J. Korean Math. Soc. 46 (2009), no. 6, 1165-1178. https://doi.org/10.4134/JKMS.2009.46.6.1165
  8. Q. Zhang, X. Li, and M. Su, Finite p-groups whose nonnormal subgroups have orders at most p3, Front. Math. China 9 (2014), no. 5, 1169-1194. https://doi.org/10.1007/s11464-014-0389-z
  9. Q. Zhang and M. Su, Finite 2-groups whose nonnormal subgroups have orders at most 23, Front. Math. China 7 (2012), no. 5, 971-1003. https://doi.org/10.1007/s11464-012-0216-3
  10. J. Zhang and Q. Zhang, Finite p-groups all of whose minimal nonnormal subgroups posses large normal closures, Comm. Algebra 44 (2016), no. 2, 628-638. https://doi.org/10.1080/00927872.2014.982819