Question

Solve (In the proof, explain each step explaining why you did that. Explain as if you were teaching a class)

1. Prove that if (G,*) is abelian, then (G/H, #) is abelian

2. If F is homomorphism from (G,*) to (G',#), then prove that e=identity in G is an element in Kernel of F and that Kernel of F is normal (you do NOT need to prove all the properties of subgroup)

3. If F is a homomorphism from (G,*) onto (G',#) then prove that IG'I divides the IGI

4. Explain how the idea of "partition" is used in the proof of lagrange's theorem

5. Prove that for each non-isomorphic Abelian group of 12, is the converse of Lagrange's theorem.

Answer 1