Definition:G-Module Homomorphism

Definition

Let $\struct {G, \cdot}$ be a group.

Let $\struct {V, \phi}$ and $\struct {W, \mu}$ be $G$-modules.

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

$\forall g \in G: \forall v \in V: \map f {\map \phi {g, v} } = \map \mu {g, \map f v}$

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.

Also see

• Definition:Homomorphism (Abstract Algebra)

• Results about $G$-module homomorphisms can be found here.

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.

Sources

• 1998: David Nelson: The Penguin Dictionary of Mathematics (2nd ed.) ... (previous) ... (next): homomorphism
• 2008: David Nelson: The Penguin Dictionary of Mathematics (4th ed.) ... (previous) ... (next): homomorphism