Representation of Degree One is Irreducible

Theorem

Let $\struct {G, \cdot}$ be a finite group.

Let $\rho: G \to \GL V$ be a linear representation of $G$ on $V$ of degree $1$.

Then $\rho$ is an irreducible linear representation.

Proof

By the definition of degree of a linear representation, it is known that $\dim V = 1$.

Let $W$ be a proper vector subspace of $V$.

It follows from Dimension of Proper Subspace is Less Than its Superspace that:

$\dim W < 1$

and hence $\dim W = 0$.

Now from Trivial Vector Space iff Zero Dimension, it follows that:

$W = \set {\mathbf 0}$

But this is not a non-trivial proper subspace of $V$.

Thus $V$ has no non-trivial proper vector subspaces.

Hence, by definition, $\rho$ is an irreducible linear representation.

$\blacksquare$