# Degree of irreducible representation divides index of center

From Groupprops

*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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## Contents

## Statement

Let be a finite group and be an algebraically closed field of characteristic zero. Let be an irreducible representation of over . Then, the degree of divides the index of the center in . (Note that since the quotient by the center equals the inner automorphism group, this is equivalent to saying that the degree of 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