Degree of irreducible representation of subgroup divides degree of irreducible representation of whole group related to it via Frobenius reciprocity

Statement
Suppose $$H$$ is a subgroup of a group $$G$$, and $$K$$ is a field. Suppose $$\varphi$$ is a finite-dimensional irreducible linear representation of $$H$$ over K and $$\psi$$ is a finite-dimensional irreducible linear representation of $$G$$ over $$K$$, such that:

$$\langle \varphi, \operatorname{Res}_H^G \psi \rangle \ne 0$$

In other words, the expression appearing on the two sides of Frobenius reciprocity is nonzero.

Then, the degree of $$\varphi$$ divides the degree of $$\psi$$.

Note that this statement does not require any assumption about $$K$$ being a splitting field or either representation being absolutely irreducible.

Applications

 * Maximum degree of irreducible representation of subgroup is less than or equal to maximum degree of irreducible representation of whole group
 * lcm of degrees of irreducible representations of subgroup divides lcm of degrees of irreducible representations of whole group