Third Isomorphism Theorem

Theorem

Groups

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.

Rings

Let $R$ be a ring.

Let:

$J, K$ be ideals of $R$

$J$ be a subset of $K$.

Then:

$(1): \quad K / J$ is an ideal of $R / J$

where $K / J$ denotes the quotient ring of $K$ by $J$

$(2): \quad \dfrac {R / J} {K / J} \cong \dfrac R K$

where $\cong$ denotes ring isomorphism.

Also known as

There is no standard numbering for the Isomorphism Theorems. Different authors use different labellings.

This particular result, for example, is also known both as the first isomorphism theorem and the second isomorphism theorem, according to source.

Also see

• Isomorphism Theorems