# Quotient Theorem for Group Epimorphisms

## Theorem

Let $\struct {G, \oplus}$ and $\struct {H, \odot}$ be groups.

Let $\phi: \struct {G, \oplus} \to \struct {H, \odot}$ be a group epimorphism.

Let $e_G$ and $e_H$ be the identities of $G$ and $H$ respectively.

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

There exists one and only one group isomorphism $\psi: G / K \to H$ satisfying:

- $\psi \circ q_K = \phi$

where $q_K$ is the quotient epimorphism from $G$ to $G / K$.

## Proof

Let $\RR_\phi$ be the equivalence on $G$ defined by $\phi$.

\(\ds \forall x \in G: \, \) | \(\ds e_G\) | \(\RR_\phi\) | \(\ds x\) | |||||||||||

\(\ds \leadstoandfrom \ \ \) | \(\ds \map \phi x\) | \(=\) | \(\ds \map \phi {e_G}\) | Definition of $\RR_\phi$ | ||||||||||

\(\ds \leadstoandfrom \ \ \) | \(\ds \map \phi x\) | \(=\) | \(\ds e_H\) | Homomorphism to Group Preserves Identity |

Thus:

- $K = \eqclass {e_G} {\RR_\phi}$

From the Quotient Theorem for Epimorphisms:

- $\RR_\phi$ is compatible with $\oplus$

Thus from Kernel is Normal Subgroup of Domain:

- $K \lhd G$

From Congruence Relation induces Normal Subgroup, $\RR_\phi$ is the equivalence defined by $K$.

Thus, again by Quotient Theorem for Epimorphisms, there is a unique epimorphism $\psi: G / K \to H$ satisfying $\psi \circ q_K = \phi$.

$\blacksquare$

## Also known as

Some sources call this the **Factor Theorem for Group Epimorphisms**.

## Sources

- 1965: Seth Warner:
*Modern Algebra*... (previous) ... (next): Chapter $\text {II}$: New Structures from Old: $\S 12$: Homomorphisms: Theorem $12.6$