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 defining ingredient::order of the defining ingredient::abelianization of $$G$$, i.e., the quotient group $$G/[G,G]$$ where $$[G,G]$$ is the defining ingredient::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.