# Degree of irreducible projective representation divides index of cyclic normal subgroup

From Groupprops

## Contents

## Statement

Suppose is a finite group, is a Cyclic normal subgroup (?) of , and is an algebraically closed field whose characteristic does not divide the order of . Suppose that is an irreducible projective representation of over .

Then, the degree of divides the index of in .

This is a numerical constraint on the Degrees of irreducible projective representations (?).

## Related facts

### Corollaries

### Ordinary representations

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

## Facts used

- Degree of irreducible projective representation divides index of abelian normal subgroup to which its cohomology class restricts trivially
- Cyclic implies Schur-trivial

## Proof

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