Third Isomorphism Theorem/Groups/Corollary

Corollary to Third Isomorphism Theorem for Groups

Let $G$ be a group.

Let $N$ be a normal subgroup of $G$.

Let $q: G \to \dfrac G N$ be the quotient epimorphism from $G$ to the quotient group $\dfrac G N$.

Let $K$ be the kernel of $q$.

Then:

$\dfrac G N \cong \dfrac {G / K} {N / K}$

Proof

From Kernel is Normal Subgroup of Domain we have that $K$ is a normal subgroup of $G$.

Thus the Third Isomorphism Theorem for Groups can be applied directly.

$\blacksquare$

Also known as

Some sources refer to this as the first isomorphism theorem.

Sources

• 1966: Richard A. Dean: Elements of Abstract Algebra ... (previous) ... (next): $\S 1.10$: Theorem $30$
• 1967: George McCarty: Topology: An Introduction with Application to Topological Groups ... (previous) ... (next): Chapter $\text{II}$: Groups: Factoring Morphisms: Corollary