Past exam of the groups and representations course of the University of Oxford / 2023 / 1 / a / Solution

A subgroup $H$ of a group $G$ is a nonempty subset closed under the inherited multiplication and inverses; equivalently, $x{y}^{-1}\in H$ whenever $x,y\in H$. A normal subgroup satisfies $gH{g}^{-1}=H$ for every $g\in G$.

A group representation on a vector space $V$ is a group homomorphism $R:G\to \mathrm{GL}(V)$, so $R(gh)=R(g)R(h)$ and $R(e)=I_V$. An irreducible representation has $V\ne 0$ and no invariant subspaces other than $0$ and $V$: there is no proper nonzero $W$ with $R(g)W\subseteq W$ for every $g\in G$. We use finite-dimensional complex representations, with continuity for Lie groups, throughout.