The purpose of this note is to give a proof of a theorem of Serre, which states that if G is a p-group which is not elementary abelian, then there exist an integer m and non-zero elements x1, …, xm ∈ H1 (G, Z/p) such that x 1 · x 2 · … · x m = 0 if p = 2 , β ( x 1 ) · β ( x 2 ) · … · β ( x m ) = 0 if p > 2 , with β the Bockstein homomorphism. Denote by mG the smallest integer m satisfying the above property. The theorem was originally proved by Serre [5], without any bound on mG. Later, in [2], Kroll showed that mG ⩽ pk − 1, with k = dimZ/pH1 (G, Z/p). Serre, in [6], also showed that mG ⩽ (pk − 1)/(p − 1). In [3], using the Evens norm map, Okuyama and Sasaki gave a proof with a slight improvement on Serre's bound; it follows from their proof (see, for example, [1, Theorem 4.7.3]) that mG ⩽ (p + 1)pk−2. However, mG can be sharpened further, as we see below. For convenience, write H*(G, Z/p) = H*(G). For every xi ∈ H1(G), set y i = { x i for p = 2 β ( x i ) for p > 2. 1991 Mathematics Subject Classification 20J06.