Find an isomorphism from Z_12 to Z_4 (direct product) Z_3. (Z_4 circle plus Z_3)
how many isomorphism are there in total.
Now if is an isomorphism then . Thus, we require that order of to be equal to . The elements that have order in are . Now confirm that in each of four cases extends to an isomorphism. Thus, that means there are four isomorphisms.
Here is another way to do this problem. Note that the number of isomorphisms between is the same as the number of automophisms of . Now use the result that . In this case .