SOME RESTRICTIONS ON NORMALIZERS OR CENTRALIZERS IN FINITE p-GROUPS

We study three restrictions on normalizers or centralizers in finite pgroups, namely: ( i) vertical bar N-G( H): H vertical bar <= 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(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.