Third Isomorphism Theorem (Groups)

Third Isomorphism Theorem (Groups). Let $G$ be a group and let $N \trianglelefteq G$ and $K \trianglelefteq G$ be normal subgroups with $N \subseteq K$. Then $K/N$ is a normal subgroup of $G/N$, and there is a canonical isomorphism of quotient groups

induced by the map $gN \mapsto gK$.

This theorem formalizes the idea that “quotienting by $N$ and then by $K/N$” is the same as quotienting directly by $K$. It can be viewed as a special case of the correspondence theorem , or proved directly using the first isomorphism theorem .

Proof sketch. Define a homomorphism $G/N \to G/K$ by $gN \mapsto gK$; it is surjective. Its kernel is precisely $K/N$. Apply the first isomorphism theorem to conclude $(G/N)/(K/N) \cong G/K$.