First Isomorphism Theorem
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.