Pls help me do these tasks, thanks. So far I just learn the basic info about iso

Question

Pls help me do these tasks, thanks. So far I just learn the basic info about isomorphism like the def of isomorphism, and I also did the task to prove any infinite cyclic groups are isomorphism which under addition. I know there are lots of proof about these task in google, but most of them include stuffs that I didnt study yet. So I really need someone can do this with good explanation. Thank you very much.

Let G be an finite cyclic group of order in with G - (a), so a has order m. Define phi : G right arrow (Zm, +m) by phi(an) = [n] for every n with 0 n o, x) defined by phi(x) = 2X for all real number x epsilon R. Show that phi is an isomorphism! Let us explore some properties of isomorphism. Let phi : (G, *) right arrow (H, ) be an isomorphism. Prove that if (G. *) is abelian, then (H, ) is also abelian.

Explanation / Answer

I have written the answer on a paper and uploaded it as several pictures at the following links:

Task- 63: Part-1: http://i.imgur.com/RWOt8pR.jpg

Part-2: http://i.imgur.com/ygHvzoV.jpg

Task-64: PArt-1: http://i.imgur.com/y6x61hd.jpg

PArt-2: http://i.imgur.com/Kxqm3Cg.jpg

Task-65: http://i.imgur.com/JnkEjB1.jpg

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