If \(M \in {}_{G}{\mathsf{Mod}}\) is an irreducible representation of \(G\) with \(\dim_k M < \infty\) and \(k=\overline{k}\), then there is an isomorphism \begin{align*} M & \xrightarrow{\sim} \mathop{\mathrm{Aut}}_G(M, M) .\end{align*}
Let \(k\) be a field with \(\operatorname{ch}(k)\) not dividing \({\sharp}G\). Then any finite-dimensional representation of \(G\) decomposes into a direct sum of irreducible representations.
The character of a representation \(M\) is the trace of the map \begin{align*} T_g: M &\to M \\ m &\mapsto g\curvearrowright m .\end{align*}