---
tag: 0112DTS464
kind: definition
name: Image of a homomorphism
msc: [20A05]
defines: 0S21GNW9ZB
uses: [0HV9Y7WG5R, 0N0PXCTYA7]
sources:
  - url: https://en.wikipedia.org/wiki/Group_homomorphism
    locator: Image and kernel
---

## Definition

Let $G$ and $H$ be [[groups|G5S1M9]] and let $\varphi \colon G \to H$ be a [[group homomorphism|0JYTG5K9Y4]].
The image of $\varphi$ is the set $\operatorname{im} \varphi = \varphi(G) = \{\varphi(g) : g \in G\}$.

## Remarks

The image is a [[subgroup|0X1JXT47GA]] of $H$, because $\varphi(a) \varphi(b)^{-1} = \varphi(ab^{-1})$.
The homomorphism $\varphi$ is surjective exactly when $\operatorname{im} \varphi = H$.
The image of the map $n \mapsto 2^n$ from $\mathbb{Z}$ to $\mathbb{Q}^\times$ is the set of integer powers of $2$.

## Explanations

### L1 human

The image is the part of the target group that the homomorphism actually reaches.
It is always a group of its own, inside the target.
The [[first isomorphism theorem|0CXXJNG5B6]] says that the image looks like a quotient of the source group.