Category:First Isomorphism Theorem

This category contains pages concerning First Isomorphism 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 $\Img T$.