Suppose $G$ is a group and that every irreducible representation of $G$ has dimension $1$. Why does this mean that $G$ is abelian?
The number of $1$-dimensional representations of $G$ is given by $|G/G'|$, where $G'$ is the derived subgroup of $G$. So if every irreducible representation of $G$ has degree $1$, then the number of conjugacy classes of $G$ is equal to $|G/G'|$. I can't see how to conclude that $G$ is abelian (or if this is the right approach).
Any help appreciated. Thanks