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

Statement
The maximum of the fact about::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.

Similar facts

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

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) uses::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).