Second Isomorphism Theorem

Theorem

Let $G$ be a group, and let:

$(1): \quad H$ be a subgroup of $G$

$(2): \quad N$ be a normal subgroup of $G$.

Then:

$\dfrac H {H \cap N} \cong \dfrac {H N} N$

where $\cong$ denotes group isomorphism.

Let $R$ be a ring, and let:

$S$ be a subring of $R$

$J$ be an ideal of $R$.

Then:

$(1): \quad S + J$ is a subring of $R$

$(2): \quad J$ is an ideal of $S + J$

$(3): \quad S \cap J$ is an ideal of $S$

$(4): \quad \dfrac S {S \cap J} \cong \dfrac {S + J} J$

where $\cong$ denotes group isomorphism.

This result is also referred to by some sources as the first isomorphism theorem.

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

This particular result, for example, is also known as the first isomorphism theorem.