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

TL;DR

This paper improves bounds on the exponents of odd degree elements in the integer cohomology ring of finite groups, showing they are tightly constrained by the group's order, and explores the properties of zero divisors using Tate cohomology duality.

Contribution

It establishes a new upper bound on the exponent of odd degree cohomology elements and introduces a duality-based approach to analyze zero divisors in the cohomology ring.

Findings

The exponent of odd degree cohomology divides 2|G| and its square divides 2|G|.

Examples demonstrate the bound on the exponent is sharp.

Zero divisors have a complementary exponent relation proven via Tate cohomology duality.

Abstract

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.