Degree of irreducible representation divides index of center

This article gives the statement, and possibly proof, of a constraint on numerical invariants that can be associated with a finite group

This article states a result of the form that one natural number divides another. Specifically, the (degree of a linear representation) of a/an/the (irreducible linear representation) divides the (index of a subgroup) of a/an/the (center).
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Statement

Let $G$ be a finite group and $k$ be an algebraically closed field of characteristic zero. Let $\rho$ be an irreducible representation of $G$ over $k$. Then, the degree of $\rho$ divides the index $[G:Z(G)]$ of the center $Z(G)$ in $G$. (Note that since the quotient by the center equals the inner automorphism group, this is equivalent to saying that the degree of $\rho$ divides the order of the inner automorphism group).

Related facts

Similar facts about degrees of irreducible representations

• Degree of irreducible representation divides group order
• Order of inner automorphism group bounds square of degree of irreducible representation
• Degree of irreducible representation divides index of abelian normal subgroup
• Sum of squares of degrees of irreducible representations equals order of group

Analogous facts about size of conjugacy class

• Size of conjugacy class divides order of inner automorphism group
• Size of conjugacy class divides order of group