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.

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- Nov 8th 2008, 01:44 PMJuancd08Isomorphism
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. - Nov 8th 2008, 02:24 PMThePerfectHacker
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 . - Nov 8th 2008, 06:48 PMJuancd08order?
how do you get that those elements have order 12