Centralizers of elementary Abelian subgroups in finite p-groups

Abstract

In this paper we show that if P is a finite p-group with the elements of order p (orders 2 and 4, ifp = 2) central, then the commutator group P’ and the central factor group P/Z(P) h ave the same exponent. We obtain new proofs and analogs for p = 2 of two recent results of van der Waall [4, Theorems 2.31. We also obtain a new proof of a result of Alperin [I, III, (12.1)] on the structure of centralizers of maximal elementary Abelian normal subgroups and an analog of his result for maximal elementary Abelian characteristic subgroups. The notation is standard (cf. [l]) with the following additions: For Q a finite p-group, we write