Number of one-dimensional representations equals order of abelianization

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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

See more facts under degrees of irreducible representations.