The cohomology of the sporadic group $J_2$ over ${\mathbb{F}}_3$

Abstract

We describe the cohomology ring $H^{\ast }(J_2;{\mathbb{F}}_3)$ both as subring of $H^{\ast }(3_{+}^{1+2};{\mathbb{F}}_3)$ and with an abstract presentation. We also give its Poincaré series. We use as tool a spectral sequence for the strongly closed $3$-subgroup of $J_2$. This method might be used to compute the cohomology of any finite simple group with a strongly closed $p$-subgroup.