Finite p-groups that determine p-nilpotency locally

Let G be a finite group, and let p be a prime number. It might happen that the p-Sylow normalizer N_G(P), P ∈ Sylp(G), of G is p-nilpotent, but G will not be p-nilpotent (see Ex. 1.1). However, under certain hypothesis on the structure of the Sylow p-subgroup P of G, this phenomenon cannot occur, e.g., by J. Tate’s p-nilpotency criterion this is the case if P is a Swan group in the sense of H-W. Henn and S. Priddy. In this note we show that if P does not contain subgroups of a certain isomorphism type Y_p(m) - in which case we call the p-group P slim - the previously mentioned phenomenon will not occur provided p is odd. For p = 2 the same remains true if P is D_8-free (see Main Theorem).