Image of a homomorphism

Definition

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

Remarks

The image is a subgroup 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 says that the image looks like a quotient of the source group.