# Number of one-dimensional representations equals order of abelianization

From Groupprops

## Statement

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

## Related facts

See more facts under degrees of irreducible representations.