First Isomorphism Theorem

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Context

This theorem applies for Groups, Rings, Modules, Algebras, and any other algebraic structure where you see the word homomorphism.

It is a categorical result; that is, it is independent of the structure used.


Theorem

Groups

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

Let $\map \ker \phi$ be the kernel of $\phi$.


Then:

$\Img \phi \cong G_1 / \map \ker \phi$

where $\cong$ denotes group isomorphism.


Rings

Let $\phi: R \to S$ be a ring homomorphism.

Let $\map \ker \phi$ be the kernel of $\phi$.


Then:

$\Img \phi \cong R / \map \ker \phi$

where $\cong$ denotes ring isomorphism.


Vector Spaces

Let $K$ be a field.

Let $X$ and $Y$ be vector spaces over $K$.

Let $T : X \to Y$ be a linear transformation.

Let $\ker T$ be the kernel of $T$.

Let $X/\ker T$ be the quotient vector space of $X$ modulo $\ker T$.


Then $X/\ker T$ is isomorphic to $\Img T$ as a vector space.


Topological Vector Spaces

Let $K$ be a topological field.

Let $\struct {X, \tau_X}$ and $\struct {Y, \tau_Y}$ be vector spaces over $K$.

Let $T : X \to Y$ be a continuous and open linear transformation.

Let $\ker T$ be the kernel of $T$.

Let $X/\ker T$ be the quotient topological vector space of $X$ modulo $\ker T$.


Then $X/\ker T$ is topologically isomorphic to $\Img T$.


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 as the Homomorphism Theorem.


Also see