Representation of Degree One is Irreducible

Theorem
Let $(G,\cdot)$ be a finite group and $\rho:G\to\operatorname{GL}(V)$ a representation of degree $1$.

Then $\rho$ is an  irreducible  representation.

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

Take $W$ a proper subspace of $V$, using Dimension of Proper Subspace Less Than its Superspace then $\dim(W)<1\Rightarrow\dim(W)=0$. Hence $W=\{0\}$; but this is not a proper subspace of $V$. Thus $V$ has no proper subspaces.

In conclusion $\rho$ is irreducible  by definition.