Exponent of the cohomology of a product of groups

Question

Suppose $G$, $H$ are finite groups and $M$ is a module over $G\times H$.

Question: Is the exponent of $H^i(G\times H,M)$ a divisor of $lcm(|G|,|H|)$ for $i> 0$ ?

The Künneth formula answers the question affirmatively if $M$ is trivial or, more generally, if one of the groups acts trivially on $M$. But I don't know what to expect if $M$ is non-trivial.

Does this follow from the LHSSS of the (trivial) extension $H\to G\times H \to G$? — Mark Grant, Apr 26 '13 at 06:33
I think Mark means the Lyndon-Hochschild-Serre spectral sequence. — Demin Hu, Apr 26 '13 at 07:00
@Mark: I also thought about the spectral sequence but since $E_\infty$ is just a subquotient of $H^i(G\times H,M)$, I don't know if there could be extension problems leading to higher exponents ? — Demin Hu, Apr 26 '13 at 07:09

score 5 · Accepted Answer · answered Apr 26 '13 at 12:00

For any finite group $\Gamma$, if $I$ is the augmentation ideal of ${\mathbb Z}\Gamma$, then $H^1(\Gamma,I)\cong {\mathbb Z}/|\Gamma|{\mathbb Z}$, which gives a counterexample if you take $\Gamma = G\times H$ for any $G$ and $H$ whose orders are not coprime.

Exponent of the cohomology of a product of groups

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