Quotient Group

Given a group $G$ and a normal subgroup $N \trianglelefteq G$, the quotient group $G/N$ is the set of cosets $\{gN \mid g \in G\}$ with operation $(gN)(hN) = (gh)N$. Normality of $N$ is what makes this operation well-defined.

The quotient group captures the structure of $G$ “up to $N$” — it identifies elements that differ only by an element of $N$. The canonical projection $\pi : G \to G/N$ defined by $\pi(g) = gN$ is a surjective group homomorphism with kernel $N$.

The first isomorphism theorem states that for any group homomorphism $\phi : G \to H$, the quotient $G/\ker(\phi)$ is isomorphic to the image of $\phi$. This theorem connects quotient groups, kernels, and images into a single coherent picture.