DOI QR코드

DOI QR Code

NOMALIZERS OF NONNORMAL SUBGROUPS OF FINITE p-GROUPS

  • Zhang, Qinhai ;
  • Gao, Juan
  • Received : 2010.10.21
  • Published : 2012.01.01

Abstract

Assume G is a finite p-group and i is a fixed positive integer. In this paper, finite p-groups G with ${\mid}N_G(H):H{\mid}=p^i$ for all nonnormal subgroups H are classified up to isomorphism. As a corollary, this answer Problem 116(i) proposed by Y. Berkovich in his book "Groups of Prime Power Order Vol. I" in 2008.

Keywords

finite p-groups;nonnormal subgroups;self-normalizer;central extensions

References

  1. L. J. An, A classification of nite meta-Hamilton p-groups, Ph. D. dissertation, Beijing University, 2009.
  2. Y. Berkovich, Groups of Prime Power Order. Vol. 1, With a foreword by Zvonimir Janko. de Gruyter Expositions in Mathematics, 46. Walter de Gruyter GmbH & Co. KG, Berlin, 2008.
  3. R. W. Carter, Nilpotent self-normalizing subgroups of soluble groups, Math. Z. 75 (1960/1961), 136-139. https://doi.org/10.1007/BF01211016
  4. A. Fattahi, Groups with only normal and abnormal subgroups, J. Algebra 28 (1974), 15-19. https://doi.org/10.1016/0021-8693(74)90019-2
  5. B. Huppert, Endliche Gruppen I, Spriger-Verlag, Berlin, Heidelberg, New York, 1967.
  6. L. L. Li, H. P. Qu, and G. Y. Chen, Central extension of inner abelian pp-groups. I, Acta Math. Sinica (Chin. Ser.) 53 (2010), no. 4, 675-684.
  7. D. S. Passman, Nonnormal subgroups of p-groups, J. Algebra 15 (1970), 352-370. https://doi.org/10.1016/0021-8693(70)90064-5
  8. L. Redei, Das schiefe Product in der Gruppentheorie, Comment. Math. Helv. 20 (1947), 225-267. https://doi.org/10.1007/BF02568131
  9. M. Y. Xu, A theorem on metabelian p-groups and some consequences, Chin. Ann. Math. Ser. B 5 (1984), no. 1, 1-6.
  10. M. Y. Xu and H. P. Qu, Finite p-Groups, Beijing University Press, Beijing, 2010.
  11. Q. H. Zhang and J. X. Wang, Finite groups with quasi-normal and self-normalizer subgroups, Acta Math. Sinica (Chin. Ser.) 38 (1995), no. 3, 381-385.
  12. Q. H. Zhang, Finite groups with only seminormal and abnormal subgroups, J. Math. Study 29 (1996), no. 4, 10-15.
  13. Q. H. Zhang, Finite groups with only ss-quasinormal and abnormal subgroups, Northeast. Math. J. 14 (1998), no. 1, 41-46.
  14. Q. H. Zhang, s-semipermutability and abnormality in finite groups, Comm. Algebra 27 (1999), no. 9, 4515-4524. https://doi.org/10.1080/00927879908826711
  15. Q. H. Zhang, X. Q. Guo, H. P. Qu, and M. Y. 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

Cited by

  1. Some restrictions on normalizers or centralizers in finite p-groups vol.208, pp.1, 2015, https://doi.org/10.1007/s11856-015-1197-1
  2. Some finiteness conditions on normalizers or centralizers in groups 2017, https://doi.org/10.1080/00927872.2017.1363223
  3. Finite $$p$$ p -Groups all of Whose Nonnormal Subgroups Posses Special Normalizers vol.40, pp.1, 2017, https://doi.org/10.1007/s40840-016-0322-6
  4. A finiteness condition on centralizers in locally nilpotent groups vol.182, pp.2, 2017, https://doi.org/10.1007/s00605-015-0854-0

Acknowledgement

Supported by : NSFC, NSF of Shanxi Province