Isomorphism Theorems

There is no standard numbering for the Isomorphism Theorems. Different authors use different labellings. Therefore, the following nomenclature is to a greater or lesser extent arbitrary.

= First Isomorphism Theorem =

Let $$\phi: G_1 \to G_2$$ be a group homomorphism.

Let $$\mathrm {ker} \left({\phi}\right)$$ be the kernel of $$\phi$$.

Then $$\mathrm {Im} \left({\phi}\right) \cong G_1 / \mathrm {ker} \left({\phi}\right)$$.

See First Isomorphism Theorem.

Some authors call this the homomorphism theorem.

Others combine this result with Kernel is Subgroup and Kernel is Normal Subgroup of Domain.

= Second Isomorphism Theorem =

Let $$H \le G, N \triangleleft G$$.

Then $$\frac H {H \cap N} \cong \frac {H N} N$$

See Second Isomorphism Theorem.

Some authors call this the first isomorphism theorem.

= Third Isomorphism Theorem =

Let $$H \triangleleft G, N \triangleleft G$$, $$N \subseteq H$$.

Then $$H / N \triangleleft G / N$$ and:

$$\frac {G / N} {H / N} \cong \frac G H$$.

See Third Isomorphism Theorem.

Some authors call this the first isomorphism theorem. Some call it the second isomorphism theorem.