Let $G$ be a finite group and $G' = [G,G]$ be its commutator subgroup, which is defined to be the subgroup generated by elements $[g,h] = g^{-1}h^{-1}gh$ for all $g,h \in G$, where $G'$ is a normal subgroup of $G$. And $G/G'$ is abelian.
Prove that for any field $\mathbb{K}$, the degree $1$ representations of $G$ over $\mathbb{K}$ are in bijection with the degree $1$ representations of $G/G'$ over $\mathbb{K}$.
attempt: Suppose $\phi_1: G → GL(\mathbb{K})$ be a representation of degree 1. is similar to $\phi_1: G → \mathbb{K}^{\times}$.
And define $\phi_2 : G/G'→ \mathbb{K}^{\times}$
Do I have to show $\phi_1 → \phi_2$ and show it's a bijective function? I am not sure what I have to show. Can someone please help? Thank you!