Let be a homomorphism. Prove that kernel of is infinite.

**Solution**

If is a finite subgroup then all must have norm , i.e. . ( Otherwise otherwise is an infinite sequence of distinct elements in . )

Suppose , that is finite and is the kernel of . Then let and

So, . But this is a contradiction since .

Hence, no finite subgroup of can be a kernel.

**Note: **The homomorphisms that can easily be described are for the form .