For a finite group $G$, let $\text{ rdim}(G)$ denote the smallest dimension of a faithful, complex linear representation of $G$. It is clear that $\text{ rdim}(H)\leq \text{ rdim}(G)$ for any subgroup $H$ of $G$. We consider $G$ with the property that $\text{ rdim}(H)<\text{ rdim}(G)$ whenever $H$ is a proper subgroup of $G$, in particular proving a classification of such groups when $G$ is abelian or $\text{ rdim}(G)\leq 3$.
Categories: math.GR