Let p be an odd prime number and let n be an arbitrary positive integer. We prove that there exists a p-group G whose mod-p cohomology ring has a nilpotent element xi is an element of H-2(G) satisfying xi(n) not equal 0, xi(n+p-1) = 0. As a corollary, we exhibit a p-group whose mod-p cohomology ring contains an element of nilpotency degree n+1.
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