Number of one-dimensional representations equals order of abelianization

Statement

Suppose $G$ is a finite group and $K$ is a splitting field for $G$. Then, the number of one-dimensional representations of $G$ over $K$ (up to equivalence of representations) equals the order of the abelianization of $G$, i.e., the quotient group $G/[G,G]$ where $[G,G]$ is the derived subgroup of $G$. In particular, it equals the index of $[G,G]$ in $G$.

Related facts

• Number of irreducible representations equals number of conjugacy classes

See more facts under degrees of irreducible representations.