Representation of Degree One is Irreducible

Theorem
Let $\left({G, \cdot}\right)$ be a finite group.

Let $\rho: G \to \operatorname{GL} \left({V}\right)$ 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 $\dim \left({V}\right) = 1$.

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

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


 * $\dim \left({W}\right) < 1$

and hence $\dim \left({W}\right) = 0$.

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


 * $W = \left\{{\mathbf 0}\right\}$

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.