# Maximum degree of irreducible real representation is at most twice maximum degree of irreducible complex representation

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## Statement

The maximum of the Degrees of irreducible representations (?) over the reals for a finite group $G$ is at most twice the maximum of the degrees of irreducible representations over the complex numbers.

## Related facts

### Combination applications

• Combining with square bound (order of inner automorphism group bounds square of degree of irreducible representation): Combined with the fact that the maximum of the degrees of irreducible representations over the complex numbers is bounded by $\sqrt{|\operatorname{Inn}(G)|}$, the maximum of the degrees of irreducible representations over the real numbers is bounded by $2\sqrt{|\operatorname{Inn}(G)|}$.

## Facts used

1. Degree of irreducible real representation either equals or is twice of degree of irreducible complex representation

## Proof

The proof is a straightforward application of fact (1).