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Let $k$ be an algebraically closed field of characteristic two and let $Q_8$be the quaternion group of order $8$. We determine the Gerstenhaber Lie algebrastructure and the Batalin-Vilkovisky structure on the Hochschild cohomologyring of the group algebra $kQ_8$.

On the Hochschild cohomology ring of the quaternion group of order eight in characteristic two