Group Homomorphisms
Definition of Homomorphism
Properties of Homomorphisms
Kernel and Image
Isomorphisms
First Isomorphism Theorem
If is a homomorphism, then the quotient group is isomorphic to .
Examples
Example 1: defined by is a surjective homomorphism with .
By the First Isomorphism Theorem: .
Example 2: is a homomorphism with .
Normal Subgroups
Second Isomorphism Theorem
If and , then .
Third Isomorphism Theorem
If and with , then .
Automorphisms
An automorphism is an isomorphism from a group to itself. The set of all automorphisms of forms a group .