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Math Help - Isomorphism in External Product

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    Isomorphism in External Product

    Is S_{3} \oplus Z_{2} \approx Z_{12}, Z_{6} \oplus Z_{2}, A_{4}, \_or\_ D_{6}

    Now, S3 is not abelian, thus rule out both Z_12 and Z_{6} \oplus Z_{2} , which are abelian.

    Then, I can't really determine the rest, they are both non-cyclic, would that help?
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    Quote Originally Posted by tttcomrader View Post
    Is S_{3} \oplus Z_{2} \approx Z_{12}, Z_{6} \oplus Z_{2}, A_{4}, \_or\_ D_{6}

    Now, S3 is not abelian, thus rule out both Z_12 and Z_{6} \oplus Z_{2} , which are abelian.

    Then, I can't really determine the rest, they are both non-cyclic, would that help?
    Prove the following simple there.

    G and H are abelian if and only if G\times H.
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  3. #3
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    Quote Originally Posted by tttcomrader View Post
    Is S_{3} \oplus Z_{2} \approx Z_{12}, Z_{6} \oplus Z_{2}, A_{4}, \_or\_ D_{6}

    Now, S3 is not abelian, thus rule out both Z_12 and Z_{6} \oplus Z_{2} , which are abelian.

    Then, I can't really determine the rest, they are both non-cyclic, would that help?
    You could look at orders of elements: for example, A_4 has eight elements of order 3 and three elements of order 2 (and the identity of order 1, of course).
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