# Schur index divides degree of irreducible representation

From Groupprops

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 is a finite group, is an Irreducible linear representation (?) over a splitting field for , is the character of , and is the Schur index (?) of . Then, divides the degree of (which is the same as the degree of ). 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 to be irreducible over a splitting field, we could instead require to be absolutely irreducible over any field whose characteristic does not divide the order of ).

In particular:

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

## Related facts

### Corollaries

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

- Combining with degree of irreducible representation divides index of abelian normal subgroup yields that Schur index of irreducible representation divides index of abelian normal subgroup.
- Order of inner automorphism group bounds square of degree of irreducible representation yields that order of inner automorphism group bounds square of Schur index of irreducible representation.