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Exponents of Zero divisors in the Cohomology ring of a finite group

It is well known that the positive degree cohomology of a finite group G is annihilated by |G|. We improve on this bound in the case of odd degree elements in the integer cohomology ring and show that $e_{odd}(G)$, the exponent of the $\oplus_{k=0}^{\infty} H^{2k+1}(G,\mathbb{Z})$ satisfies $e_{odd}(G)^2$ divides 2|G| and in particular $e_{odd}(G) \leq \sqrt{2|G|}.$ We also provide examples to show this bound for $e_{odd}(G)$ is sharp as a general bound over all finite groups G. The result comes from a fact about zero divisors having "complementary exponent" which we prove using duality in Tate cohomology. More particularly if $α, β$ are elements of positive degree in $H^*(G,\mathbb{Z})$ satisfying $αβ= 0$ then the order of $β$, $o(β)$ divides $\frac{|G|}{o(α)}$. We also apply this fact to get some results on elements of exceptionally high exponent in the cohomology ring.