Let $G$ be a finite group with faithful irreducible representation $\gamma: G \to GL\_n(\mathbb{C})$, $n>1$.


Can we put a bound on the size of $G$? What if $G$ is nilpotent?



Bound on the size of a group given a faithful irrep of a certain dimension
Jordan's theorem says that there is a function $f : \mathbb{N} \to \mathbb{N}$ such that whenever $G$ is a finite subgroup of ${\rm GL}(n,\mathbb{C}),$ there is an Abelian onrmal subgroup $A$ such that $[G:A] \leq f(n)$. Explicit bounds were given later, which can be much improved by invoking the classification of finite simple groups. If $G$ is a primitive subgroup of ${\rm SL}(n,\mathbb{C})$, then it follows from Jordan's theorem that $|G|$ is bounded in terms of $n$. Recall that a primitive representation is an irreducible one which is not equivalent to a representation induced from a representation of a proper subgroup.


