Pls help me with these 2 tasks, thanks very much, Let phi: (G) rightarrow (H, ph

Question

Pls help me with these 2 tasks, thanks very much,

Let phi: (G) rightarrow (H, phi) be an isomorphism. Prove that if (G,*) is cyclic, then (H,0) is also cyclic Let phi: (G,*) rihtarrow (H, phi) be an isomorphism. Prove that if (G,*) has an element of order n then(H, phi) also has an element of order n. How does one recognize when two groups are not isomorphic? By definition, We have to show that there is no isomorphism between the two groups but in practice, this is not possible because you have to go through all the possible functions between the two groups and show that they are not isomorphism's. It is more practical to do one of the following: Show that the two groups have different order (or different cardinality). Show that one group is a beldam and the other is not. Show that one group is cyclic and the other is not. Show that one group has an element of order n and the other does not have one. This list is by no means exhaustive: it merely illustrates the Kind of things to be on the lookout for.

Explanation / Answer

I have written the answer on a paper and uploaded it as 4 pictures at the following links:

Task-66 : Part-1: http://i.imgur.com/8p4nWPJ.jpg

Part-2: http://i.imgur.com/6THP1Mx.jpg

Task-67: Part-1: http://i.imgur.com/BdJjIpF.jpg

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

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