5.11 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.

5.11.1 Group homomorphisms

Suppose (G, ○) and (H, ×) are two groups, which we first explored in subsection 3.6.1f: G → H is a group homomorphism if for any two elements a and b in G,

f(a ○ b) = f(a) × f(b).

This means that f is not just a function, but it preserves the operations on 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⁻¹), which means f(a⁻¹) = f(a)⁻¹.

The set of all elements a in G such that f(a) = id_H is called the kernel of f. It is a subgroup of G.

If...