# Group Homomorphism

By homomorphism we mean a mapping from one algebraic system to a like algebraic system which preserves structures.

__Definition__

Let and be any two groups with binary operation and respectively. Then a mapping is said to be a homomorphism if for all ,

A homomorphism , which at the same time is also onto is said to be an epimorphism.

A homomorphism , which at the same time is also one-one is said to be an monomorphism.

A group is called a homomorphism image of a group , if there exist a homomorphism of onto . A homomorphism of a group into itself is called an edomorphism.

__Examples__**:**

**(i) **Let be any group under binary operation . If for every then is a homomorphism because

**(ii)** Let be the group of integers under addition, let be the group of integers under addition modulo . If be defined by remainder of on division by , then this is a homomorphism.

**(iii)** Let be any group under addition. If , then the mapping , is a homomorphism because for all , and , so that

**(iv)** Let be the group of integers under addition and let . If for all , , then is a homomorphism because