Some restrictions on normalizers or centralizers in finite p-groups

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():|≤p^k for every non-normal in G, and (iii) |C_G(g) : | ≤ p^k for every 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. (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| ≤ 2^f(k) for some function f(k) depending only on k.