# Homomorphism and Isomorphism

• ### 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 […]

• ### Kernel of Homomorphism

Definition If is a homomorphism of a group into a , then the set of all those elements of which is mapped by onto the identity of is called the kernel of the homomorphism . Theorem: Let and be any two groups and let and be their respective identities. If is a homomorphism of into […]

• ### Examples of Group Homomorphism

Here’s some examples about the concept of group Homomorphism, as follows:  Example 1: Let which forms a group under multiplication and the group of all integers under addition, prove that the mapping from onto such that is a homomorphism. Solution: Since , for all Hence is a homomorphism. Example 2: Show that the mapping of […]

• ### Group Isomorphism

Definition Let and be any two groups with binary operation and respectively. If there exist a one-one onto mapping such that Then the group is said to be isomorphic to the group , and the mapping is said to be an isomorphism. If is isomorphic to , we write or .   In other words, […]

• ### Properties of Isomorphism

Theorem 1: If isomorphism exists between two groups, then the identities corresponds, i.e. if is an isomorphism and are respectively the identities in , then . Theorem 2: If isomorphism exists between two groups, then the identities corresponds, i.e. if is an isomorphism and , where then . Theorem 3: In an isomorphism the order […]

• ### Isomorphism of Cyclic Groups

Theorem 1: Cyclic group of same order are isomorphic. Proof: Let andbe two cyclic groups of order , which are generated by and respectively. Then and The mapping , defined by , is isomorphism. For, Therefore the groups are isomorphic.     Theorem 2: An infinite cyclic group is isomorphic to the additive group of integers. […]

• ### Cayley's Theorem

Cayley’s Theorem: Every group is isomorphic to a permutation group. Proof: Let be a finite groups of order . If , then , . Now consider a function from into defined by For Therefore, function is one-one. The function is also onto because if is any element of  then there exist an element such that […]

• ### Examples of Group Isomorphism

Example 1: Show that the multiplicative group consisting of three cube roots of unity is isomorphic to the group of residue classes under addition of residue classes . Solution: Let us consider the composition tables of two structures as given below:   From this table it is evident that if are replaced by respectively in […]