Recently I've been revising a course on Representation Theory. As part of that course, we ended up proving that the intersection of the kernels of all 1D representations of a group is its derived group, i.e. the group generated by its collection of commutators.
Let's now say that $G^{(n)}$ is intersection of the kernels of all the group homomorphisms $G\to \text{GL}(\mathbb{C}^n)$ (so the above result tells us $G^{(1)} = \delta(G)$). Is there much known about these $G^{(n)}$?
One quick result is that for any finite group, $G^{(n)}$ is trivial for all sufficiently large $n$. What else can we say?