# Definition:G-Module Homomorphism

## Definition

Let $\left({G, \cdot}\right)$ be a group.

Let $\left({V, \phi}\right)$ and $\left({W, \mu}\right)$ be $G$-modules.

Then a linear transformation $f: V \to W$ is called a $G$-module homomorphism if and only if:

$\forall g \in G: \forall v \in V: f \left({\phi \left({g, v}\right)}\right) = \mu \left({g, f \left({v}\right)}\right)$

## Also known as

Group theorists commonly refer to a $G$-module homomorphism as a $G$-intertwining map or simply an intertwining map.

Also the term $G$-equivariant map can be seen.

## 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.