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.
If is a homomorphism then (where ).
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.
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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 .