# Schur index divides degree of irreducible representation

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

Suppose $G$ is a finite group, $\varphi$ is an Irreducible linear representation (?) over a splitting field for $G$, $\chi$ is the character of $\varphi$, and $m(\chi)$ is the Schur index (?) of $\chi$. Then, $m(\chi)$ divides the degree of $\varphi$ (which is the same as the degree of $\chi$). In particular, this bounds the possible values of the Schur index for irreducible representations in terms of the Degrees of irreducible representations (?).

(Note that instead of requiring $\varphi$ to be irreducible over a splitting field, we could instead require $\varphi$ to be absolutely irreducible over any field whose characteristic does not divide the order of $G$).

In particular:

• The Schur index of a linear character is $1$ (this is obvious even otherwise).
• The Schur index of an irreducible character of prime degree is either $1$ or equal to that prime.

## Related facts

### Corollaries

This statement can be combined with results about the degrees of irreducible representations. In particular: