Maximum degree of irreducible representation of group is less than or equal to product of maximum degree of irreducible representation of subgroup and index of subgroup

Statement
Suppose $$G$$ is a finite group, $$H$$ is a subgroup, and $$K$$ is a field whose characteristic does not divide the order of $$G$$ (we do not require $$K$$ to be a splitting field, though the splitting field case is of particular interest).

Then, the fact about::maximum degree of irreducible representation of $$G$$ over $$K$$ is less than or equal to the product:

(maximum degree of irreducible representation of $$H$$ over $$K$$) times (index of $$H$$ in $$G$$, i.e., $$[G:H]$$)

Facts used

 * 1) uses::Orthogonal projection formula
 * 2) uses::Frobenius reciprocity

Related facts

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

Proof in characteristic zero
Given: $$G$$ is a finite group, $$H$$ is a subgroup, and $$K$$ is a field of characteristic zero. $$H$$ has index $$d$$ in $$G$$. $$m$$ is the maximum of the degrees of irreducible representations of $$G$$ over $$K$$.

To prove: $$H$$ has an irreducible representation over $$K$$ whose degree is at least $$m/d$$.

Proof: