Third Isomorphism Theorem/Groups

Theorem
Let $G$ be a group, and let:


 * $H, N$ be normal subgroups of $G$
 * $N$ be a subset of $H$.

Then:
 * $(1): \quad N$ is a normal subgroup of $H$


 * $(2): \quad H / N$ is a normal subgroup of $G / N$
 * where $H / N$ denotes the quotient group of $H$ by $N$


 * $(3): \quad \dfrac {G / N} {H / N} \cong \dfrac G H$
 * where $\cong$ denotes group isomorphism.

Also known as
This result is also referred to by some sources as the first isomorphism theorem, and by others as the second isomorphism theorem.

Also see

 * Isomorphism Theorems