# Third Isomorphism Theorem

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## 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:

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.