= Solution
A <subgroup> $H$ of a <group> $G$ is a nonempty subset closed under the inherited multiplication and inverses; equivalently, $xy^{-1}\in H$ whenever $x,y\in H$. A <normal subgroup> satisfies $gHg^{-1}=H$ for every $g\in G$.
A <group representation> on a <vector space> $V$ is a <group homomorphism> $R:G\to\operatorname{GL}(V)$, so $R(gh)=R(g)R(h)$ and $R(e)=I_V$. An <irreducible representation> has $V\ne0$ 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.
Back to article page
