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Let $k$ be a field of characteristic different from $2$.

Let $n\geq 1$ be an integer, and let $T$ be the maximal torus of the $k$-algebraic group $PGL_n$, namely the quotient of diagonal matrices by the diagonal action of $\mathbb{G}_{m,k}$.

Then let the symmetric group $S_n$ act on T by permuting the diagonal entries.

We know from general results that $H^2(S_n,T)$ is killed by $n!$, but can we do better ?

Is it killed by $n$, for example ? what is the exponent of $H^2(S_n,T)$?

Thanks in advance!

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