# Degree of irreducible representation of nontrivial finite group is strictly less than order of group

## Statement

Suppose is a nontrivial finite group of order and is a field. Then, any irreducible representation of over has degree at most .

Equality is attained for infinitely many groups: namely, where is a group of prime order and is , the field of rational numbers.

## Proof

**Given**: A group of order . A nonzero vector space , a homomorphism such that has no proper nonzero -invariant subspace.

**To prove**: The dimension of is at most .

**Proof**: Let . Consider the set:

The vector space spanned by is -invariant and nonzero, hence must equal . Since it has a spanning set of size , it has dimension at most . Further, the dimension equals if and only if is linearly independent. However, if is linearly independent, the element:

is -invariant and nonzero, hence it spans a nonzero -invariant one-dimensional subspace of . The subspace is proper since , contradicting the irreducibility assumption. Thus, cannot be linearly independent, and the dimension of is at most .