Let p be a prime number and let G be a p-group which is not elementary abelian. For every integral cohomology class ξ of G which restricts trivially to all proper subgroups, we show that ξ~p = 0 if p > 2 or deg(ξ) is even, and ξ~3 = 0 if p = 2 and deg(ξ) is odd. This result is applied to get an upper bound, which is |G|/p, for the nilpotence degrees of nilpotent integral cohomology classes of G.
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