# Hokkaido Mathematical Journal

## Hokkaido Mathematical Journal, 41 (2012) pp.11-29

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### Abstract

Let G be a finite group, and let p be a prime number. It might happen that the p-Sylow normalizer NG(P), P ∈ Sylp(G), of G is p-nilpotent, but G will not be p-nilpotent (see Example 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 Yp(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 D8-free (see Main Theorem).

MSC(Primary) 20D20 20D15 finite groups, p-nilpotency, slim p-groups, Sylow subgroups, p-nilpotent Sylow normalizer