# Examples of Group Homomorphism

Here’s some examples of the concept of group homomorphism.

__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 the symmetric group onto the multiplicative group defined by or .

According as is an even or odd permutation in is a homomorphism of onto .

__Solution__**:** We know that the product of two permutations both even or both odd is even while the product of one even and one odd permutation is odd. We shall show that

**(i)** if are both even, then

**(ii)** if are both odd, then

**(iii)** if is odd and is even, then

**(iv)** if is even and is odd, then

Thus . Also obviously is onto .

Therefore is a homomorphism of onto .

__Example 3__**:**

Show that a homomorphism from s simple group is either trivial or one-to-one.

__Solution__**:** Let be a simple group and be a homomorphism of into another group . Then the kernel is a normal subgroup of . But the only normal subgroups of the simple group are and . Hence either or . If , the image of each element of is the identity of , as such the homomorphism is trivial one. If , the homomorphism is one-to-one.