Let and be homomorpisms. Is the set a subgroup of ? Prove or disprove your answer.Problem:

It is.Claim:

Let us verify the four neccessary conditions to be a subgroupProof:

1. Clearly will inherit 's associativity.

2. You should know that . Thus .

3. Now suppose that , then , so that which of course implies .

4. Suppose then and . Therefore . Therefore

This completes the proof.

Remark:I always find the direct satsifaction of the subgroup axioms to be more instructive. You may (and should) attempt to redo this using the condition that if .