# Definition:Homomorphism (Abstract Algebra)/Image

## Definition

Let $S$ and $T$ be algebraic structures.

Let $\phi: S \to T$ be a homomorphism from $S$ to $T$.

As a homomorphism is a mapping, the homomorphic image of $\phi$ is defined in the same way as the image of a mapping:

$\Img \phi = \set {t \in T: \exists s \in S: t = \map \phi s}$

## Linguistic Note

The word homomorphism comes from the Greek morphe (μορφή) meaning form or structure, with the prefix homo- meaning similar.

Thus homomorphism means similar structure.