5.12 Homomorphisms

When functions operate on collections with algebraic structure, we usually require additional properties to be preserved. We can now redefine linear maps and transformations of vector spaces in terms of these functions, called homomorphisms. homomorphism

5.12.1 Group homomorphisms

Suppose (G, ★) and (H, •) are groups, which we first explored in section 3.6.1. The function f : G → H is a group homomorphism if for any two elements a and b in G, homomorphism$group group$homomorphism

This means that f is not just a function, but it preserves the operations of the groups.

We have the following properties for group homomorphisms:

• f (id_G) = f (id_G ★ id_G) = f (id_G) • f (id_G), which means f (id_G) = id_H.
• id_H = f (id_G) = f (a ★ a^–1) = f (a) • f (a^–...