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