  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
$\alpha, \beta$ are elements of positive degree in $H^*(G,\mathbb{Z})$
satisfying $\alpha \beta = 0$ then the order of $\beta$, $o(\beta)$ divides
$\frac{|G|}{o(\alpha)}$.
  We also apply this fact to get some results on elements of exceptionally high
exponent in the cohomology ring.
