Some restrictions on normalizers or centralizers in finite p-groups
Group Theory
2013-11-13 v1
Abstract
We study three restrictions on normalizers or centralizers in finite p-groups, namely: (i) |N_G(H) : H| <= p^k for every H non-normal in G, (ii) |N_G(<g>) : <g>| <= p^k for every <g> non-normal in G, and (iii) |C_G(g) : <g>| <= p^k for every <g> non-normal in 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. 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| <= 2^{f(k)} for some function f(k) depending only on k.
Cite
@article{arxiv.1311.2608,
title = {Some restrictions on normalizers or centralizers in finite p-groups},
author = {Gustavo A. Fernandez-Alcober and Leire Legarreta and Antonio Tortora and Maria Tota},
journal= {arXiv preprint arXiv:1311.2608},
year = {2013}
}