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Math Help - Isomorphism between D4 and Z2 x Z4

  1. #1
    Super Member Bernhard's Avatar
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    Isomorphism between D4 and Z2 x Z4

    Can anyone please help with the following problem:

    Show that  D_4 and \mathbb{Z}_2 x \mathbb{Z}_4 are not isomporphic

    Be grateful for help with the approach to this problem.

    For those who need an update or reaquaintance with  D_4 I have attached the relevant pages of Fraleigh: A First Course in Abstract Algebra.

    Peter
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  2. #2
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    Re: Isomorphism between D4 and Z2 x Z4

    Quote Originally Posted by Bernhard View Post
    Can anyone please help with the following problem:

    Show that  D_4 and \mathbb{Z}_2 x \mathbb{Z}_4 are not isomporphic

    Be grateful for help with the approach to this problem.

    For those who need an update or reaquaintance with  D_4 I have attached the relevant pages of Fraleigh: A First Course in Abstract Algebra.

    Peter
    It is known - or easy to prove - that homomorphic image of an abelian group is an abelian group, too.

    Notice that D_4 is not abelian while \mathbb{Z}_2 \times\mathbb{Z}_4 is abelian.
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  3. #3
    MHF Contributor Swlabr's Avatar
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    Re: Isomorphism between D4 and Z2 x Z4

    Quote Originally Posted by zoek View Post
    It is known - or easy to prove - that homomorphic image of an abelian group is an abelian group, too.

    Notice that D_4 is not abelian while \mathbb{Z}_2 \times\mathbb{Z}_4 is abelian.
    Another approach would be to note that that isomorphisms are bijections. Therefore, they preserve order. So, \mathbb{Z}_2\times \mathbb{Z}_4 has order 8, while the Klein 4-group has order 4. So...
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  4. #4
    Super Member Bernhard's Avatar
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    Re: Isomorphism between D4 and Z2 x Z4

    Thanks for the post

    But Swlabr. this post concerns  D_4 which has order 8 ... not the Klein 4-group (order 4) ...

    Peter
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  5. #5
    MHF Contributor Swlabr's Avatar
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    Re: Isomorphism between D4 and Z2 x Z4

    Quote Originally Posted by Bernhard View Post
    Thanks for the post

    But Swlabr. this post concerns  D_4 which has order 8 ... not the Klein 4-group (order 4) ...

    Peter
    Ah, sorry, there are two conventions - one can use either D_{n} or D_{2n} for the Dihedral group of order 2n. I should have realised from context which one you were using...
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  6. #6
    Super Member Bernhard's Avatar
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    Re: Isomorphism between D4 and Z2 x Z4

    oh ... understand!

    Peter
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