Abstract
We study three restrictions on normalizers or centralizers in finite p-groups, namely: (i) |N G (H):H|≤p k for every H \( \triangleleft / - \) G, (ii) |N G (〈g〉):〈g〉|≤p k for every 〈g〉\( \triangleleft / - \) G, and (iii) |C G (g): 〈g〉 ≤ p k for every 〈g〉 \( \triangleleft / - \) G. We prove that (i) and (ii) are equivalent, and that the order of a non-Dedekind finite p-group satisfying any of these three conditions is bounded for p > 2. (For condition (i) this fact was proved earlier by Zhang and Guo [14].) More precisely, we get the best possible bound for the order of G in all three cases, which is |G| ≤ p 2k+2. The order of the group cannot be bounded for p = 2, but we are able to identify two infinite families of 2-groups out of which |G| ≤ 2f(k) for some function f(k) depending only on k.
Similar content being viewed by others
References
T. Andreescu and D. Andrica, Number Theory: Structures, Examples, and Problems, Birkhäuser, Boston, MA, 2009.
R. Baer, Situation der Untergruppen und Struktur der Gruppe, Sitzungsberichte der Heidelberger Akademie der Wissenschaften 2 (1933), 12–17.
Y. Berkovich, Groups of Prime Power Order, Volume 1, de Gruyter Expositions in Mathematics, Vol. 46, Walter de Gruyter, Berlin, 2008.
N. Blackburn, Finite groups in which the nonnormal subgroups have nontrivial intersection, Journal of Algebra 3 (1966), 30–37.
R. Dedekind, Ueber Gruppen, deren sämmtliche Theiler Normaltheiler sind, Mathematische Annalen 48 (1897), 548–561.
G. A. Fernández-Alcober, L. Legarreta, A. Tortora and M. Tota, A restriction on centralizers in finite groups, Journal of Algebra 400 (2014), 33–42.
J. González-Sánchez and T. S. Weigel, Finite p-central groups of height k, Israel Journal of Mathematics 181 (2011), 125–143.
B. Huppert, Endliche Gruppen. I, Die Grundlehren der Mathematischen Wissenschaften, Vol. 134, Springer, Berlin-New York, 1967.
H. Kurzweil and B. Stellmacher, The Theory of Finite Groups, Universitext, Springer-Verlag, New York, 2004.
B. H. Neumann, Groups with finite classes of conjugate subgroups, Mathematische Zeitschrift 63 (1955), 76–96.
M. Suzuki, Group Theory. I, Grundlehren der Mathematischen Wissenschaften, Vol. 247, Springer-Verlag, Berlin-New York, 1982.
H. Zassenhaus, The Theory of Groups, second edition, Chelsea, New York, 1958.
Q. Zhang and J. Gao, Normalizers of nonnormal subgroups of finite p-groups, Journal of the Korean Mathematical Society 49 (2012), 201–221.
X. Zhang and X. Guo, Finite p-groups whose non-normal cyclic subgroups have small index in their normalizers, Journal of Group Theory 15 (2012), 641–659.
Author information
Authors and Affiliations
Corresponding author
Additional information
The first two authors are supported by the Spanish Government, grant MTM2011-28229-C02-02, and by the Basque Government, grants IT460-10 and IT753-13.
The last two authors would like to thank the Department of Mathematics at the University of the Basque Country for its excellent hospitality while part of this paper was being written. They also wish to thank G.N.S.A.G.A. (INdAM) for financial support.
Rights and permissions
About this article
Cite this article
Fernández-Alcober, G.A., Legarreta, L., Tortora, A. et al. Some restrictions on normalizers or centralizers in finite p-groups. Isr. J. Math. 208, 193–217 (2015). https://doi.org/10.1007/s11856-015-1197-1
Received:
Revised:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s11856-015-1197-1