**Group Homomorphisms**

Definitions, examples, and proofs talking about group homomorphisms are given.

**Homomorphisms**

A homomorphism is a function between two groups. Its a way to compare two groups for structural similarities. Homomorphisms are a powerful tool

for studying and cataloging groups.

**Isomorphisms**

An isomorphism is a homomorphism that is also a bijection. If there is an isomorphism between two groups G and H, then they are equivalent and we

say they are “isomorphic.” The groups may look different from each other, but their group properties will be the same.

**The Kernel of a Group Homomorphism**

The kernel of a group homomorphism measures how far off it is from being one-to-one (an injection). Suppose you have a group homomorphism f:G ?

H. The kernel is the set of all elements in G which map to the identity element in H. It is a subgroup in G and it depends on f. Different

homomorphisms between G and H can give different kernels.