Degree of irreducible projective representation divides index of cyclic normal subgroup

Statement
Suppose $$G$$ is a finite group, $$H$$ is a fact about::cyclic normal subgroup of $$G$$, and $$K$$ is an algebraically closed field whose characteristic does not divide the order of $$G$$. Suppose that $$\varphi$$ is an irreducible projective representation of $$G$$ over $$K$$.

Then, the degree of $$\varphi$$ divides the index of $$H$$ in $$G$$.

This is a numerical constraint on the fact about::degrees of irreducible projective representations.

Corollaries

 * Degree of irreducible projective representation divides order of group

Ordinary representations

 * Degree of irreducible representation divides index of abelian normal subgroup -- the version for ordinary (linear) representations.

Facts used

 * 1) uses::Degree of irreducible projective representation divides index of abelian normal subgroup to which its cohomology class restricts trivially
 * 2) uses::Cyclic implies Schur-trivial

Proof
The proof follows directly from Facts (1) and (2).