REMARKS ON COHOMOLOGY RINGS OF THE QUATERNION GROUP AND THE QUATERNION ALGEBRA

Abstract

We will give a homomorphism of the cohomology rings $H^{*}\left(\Gamma ,\Gamma \right)\to H^{*}\left(Q_{8,\psi }\Gamma \right)$induced by the ring homomorphism from the integral group algebra $\Lambda =\mathbb{Z}{Q}_8$ of the quaternion group $Q_8$ to the quaternion algebra $\Gamma =\Lambda e$ for a central idempotent $e$ in $\mathbb{Q}{Q}_8$.