Can someone help me with 18.13? Use the mapping theta([a] sub 6)=([a] sub 2, [a]

Question

Can someone help me with 18.13? Use the mapping theta([a] sub 6)=([a] sub 2, [a] sub 3) to show that Z6 is isomorphic to Z2 X Z3. First verify that theta is well defined.

Explanation / Answer

18.13. I am assuming you mean isomorphic (rather than equiavalent) and that q:Z6 -> Z2xZ3 is defined by q([a]6)=([a]2, [a]3). I will only prove that q is onto; perhaps that will help you see what is to be done. Compute: q([0]6)=([0]2, [0]3), q([1]6)=([1]2, [1]3), q([2]6)=([2]2, [2]3)=([0]2, [2]3), q([3]6)=([3]2, [3]3)=([1]2, [0]3) ... etc going through all of the elements of Z6 and checking that the images will cover all of the elements of Z2xZ3. do rate 5

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