Degree of irreducible representation over field of characteristic coprime to order divides product of order and Euler totient function of exponent

Statement
Suppose $$G$$ is a finite group, $$k$$ is a field whose characteristic is relatively prime to the order of $$G$$, and $$\varphi$$ is an irreducible representation of $$G$$ over $$k$$ (which need not be absolutely irreducible, i.e., it may split over some algebraic extension of $$k$$). Then, the degree of $$\varphi$$ divides the product of the order of $$G$$ and the Euler totient function of the exponent of $$G$$.

Other facts in general

 * Degree of irreducible representation of nontrivial finite group is strictly less than order of group
 * Maximum degree of irreducible real representation is at most twice maximum degree of irreducible complex representation

Facts over a splitting field
We can obtain much better bounds on the degrees of irreducible representations if $$K$$ is a splitting field for $$G$$. Specifically:


 * Degree of irreducible representation divides group order
 * Degree of irreducible representation divides order of inner automorphism group
 * Degree of irreducible representation divides index of abelian normal subgroup
 * Order of inner automorphism group bounds square of degree of irreducible representation
 * Sum of squares of degrees of irreducible representations equals order of group
 * Number of irreducible representations equals number of conjugacy classes

Facts used

 * 1) uses::Sufficiently large implies splitting: Any sufficiently large field for a finite group $$G$$, i.e., a field that contains all the $$m^{th}$$ roots of unity where $$m$$ is the exponent of $$G$$, is also a splitting field for $$G$$.
 * 2) uses::Degree of irreducible representation divides group order