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 G , (ii) | N G (〈 g 〉):〈 g 〉|≤ p k for every 〈 g 〉 G , and (iii) | C G ( g ): 〈 g 〉 ≤ p k for every 〈 g 〉 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 2 k +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 .