Equivalence classes of degree-one representations are exactly homomorphisms $G\to k^{\times }$; equivalently they factor through $G\mathrm{/}{G}^{\mathrm{\prime }}$, and they form an abelian group

Let $\rho :G\to \mathrm{GL}(V)$ be a degree-one representation. By [L1], choose a basis vector $v$ of the one-dimensional space $V$. For each $g\in G$ there is a unique scalar $\chi (g)\in k$ with $\rho (g)v=\chi (g)v$, and because $\rho (g)$ is invertible, that scalar lies in $k^{\times }$.

The representation law gives $\chi (gh)v=\rho (gh)v=\rho (g)\rho (h)v=\chi (g)\chi (h)v$, so $\chi (gh)=\chi (g)\chi (h)$. Thus $\chi :G\to k^{\times }$ is a group homomorphism. Under the identification $V\cong k$ sending $v$ to $1$, the representation becomes the normalized action $g\cdot a=\chi (g)a$ on the one-dimensional space $k$.

Conversely, if $\chi :G\to k^{\times }$ is a group homomorphism, define ${\rho }_{\chi }(g):k\to k$ by ${\rho }_{\chi }(g)(a)=\chi (g)a$. Each ${\rho }_{\chi }(g)$ is invertible with inverse multiplication by $\chi (g{)}^{-1}$, and the homomorphism law for $\chi$ gives ${\rho }_{\chi }(gh)={\rho }_{\chi }(g){\rho }_{\chi }(h)$. So ${\rho }_{\chi }$ is a degree-one representation.

If two degree-one representations are equivalent, an invertible intertwiner sends a chosen basis vector of one space to a nonzero scalar multiple of a basis vector of the other, so the character extracted in step 2.1 is unchanged; conversely, if two degree-one representations produce the same homomorphism $\chi$, then step 2.1 identifies each of them with the same normalized representation ${\rho }_{\chi }$ of step 3.1, so they are equivalent by [L4]. Therefore equivalence classes of degree-one representations are in bijection with homomorphisms $G\to k^{\times }$.

Because $k^{\times }$ is abelian by [L2], [L3] identifies the same equivalence classes with the homomorphisms factoring through $G\mathrm{/}{G}^{\mathrm{\prime }}$. If $\chi ,\psi :G\to k^{\times }$ are homomorphisms, then $(\chi \psi )(g):=\chi (g)\psi (g)$ and ${\chi }^{-1}(g):=\chi (g{)}^{-1}$ are again homomorphisms. Thus the equivalence classes of degree-one representations form an abelian group under pointwise multiplication and inversion of their corresponding homomorphisms.