By homomorphism we mean a mapping from one algebraic system with a like algebraic system which preserves structures.
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 exists a homomorphism of onto . A homomorphism of a group into itself is called an edomorphism.
(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