Abelian implies every irreducible representation is one-dimensional

Statement

For an abelian group, any irreducible representation over a splitting field is one-dimensional.

In particular, for a finite abelian group of order $n$, the Degrees of irreducible representations (?) over a splitting field are $1,1,1,\dots ,1$ and there are $n$ of them.

Related specific information

• Linear representation theory of cyclic groups
• Linear representation theory of finite abelian groups

Related facts

Generalizations

• Degree of irreducible representation divides index of abelian normal subgroup
• Degree of irreducible representation is bounded by index of abelian subgroup
• Number of one-dimensional representations equals order of abelianization