Having seen that the kernel records exactly which elements a homomorphism sends to the identity, we now ask when a homomorphism can be rebuilt from a quotient group. The focus keyword factorization through quotients means that a homomorphism $f:G\to G_1$ may pass through $G/H$ whenever $H$ is contained in the kernel of $f$. In that situation, all elements of the same coset of $H$ have the same image under $f$. The quotient has already collapsed $H$, so the new map from $G/H$ to $G_1$ is well-defined. This lesson prepares the first isomorphism theorem by explaining the exact mechanism behind quotient factorization.

Let $f:G\to G_1$ be a homomorphism and let $H\trianglelefteq G$. We say that $f$ \textbf{factors through the quotient} $G/H$ if there exists a homomorphism $\overline{f}:G/H\to G_1$ such that $$\begin{equation*} f=\overline{f}\circ \pi, \end{equation*}$$ where $\pi:G\to G/H$ is the natural homomorphism defined by $\pi(a)=aH$.