---
tag: 0GJ83E78A5
kind: statement
role: theorem
name: Universal property of the quotient group
aliases: [factorization-through-the-quotient]
msc: [20A05]
uses: [0YYWCNTPWR, 0HV9Y7WG5R, 0Z424HK8CQ]
sources:
  - url: https://en.wikipedia.org/wiki/Quotient_group
    locator: Properties
---

## Statement

Let $G$ and $H$ be [[groups|G5S1M9]], let $N$ be a [[normal subgroup|0CB0DX5BVV]] of $G$, and let $\pi \colon G \to G/N$ be the projection $\pi(g) = gN$ to the [[quotient group|0KYX2N69J0]].
Let $\varphi \colon G \to H$ be a [[group homomorphism|0JYTG5K9Y4]] with $N \subseteq \ker \varphi$.
Then there is exactly one group homomorphism $\bar{\varphi} \colon G/N \to H$ with $\bar{\varphi} \circ \pi = \varphi$, and it is given by $\bar{\varphi}(gN) = \varphi(g)$.

## Remarks

The map $\bar{\varphi}$ is well defined because $\varphi$ is constant on each coset of $N$.
The image of $\bar{\varphi}$ is the image of $\varphi$, and the kernel of $\bar{\varphi}$ is $(\ker \varphi)/N$.
With $N = \ker \varphi$, this gives the [[first isomorphism theorem|0CXXJNG5B6]].

## Explanations

### L1 human

A homomorphism that sends all of $N$ to the identity cannot tell apart elements of the same coset of $N$.
So it already makes sense on the cosets, that is, on the quotient group.
This is the standard way to define a homomorphism out of a quotient group.