This question came to me when looking at irreducible representations of a finite group over the complex numbers.
We know by Artin-Wedderburn theory that the number of irreducible complex representations of a finite group equals the number of conjugacy classes. We also know that we have a $1:1$ correspondence between one dimensional representations of $G$ and one dimensional representations of $G^{ab}$. Then it must follow that $|G^{ab}| \leq |\{\text{conjugacy classes of } G\}|$? since every one dimensional representation is trivially irreducible. If this is in fact true, then possibly there is a much simpler way requiring less machinery?