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# Group homomorphism with an infinite kernel

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 .

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