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Math Help - H normal in G

  1. #1
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    H normal in G

    group G = Z_4 x Z_4

    Let H be a subgroup of G of order 4. Explain why H is normal in G and why the quotient group of G by H is Abelian and of order 4.
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  2. #2
    MHF Contributor Drexel28's Avatar
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    Quote Originally Posted by tiasum2 View Post
    group G = Z_4 x Z_4

    Let H be a subgroup of G of order 4. Explain why H is normal in G and why the quotient group of G by H is Abelian and of order 4.
    Hint:
    Spoiler:


    It's normal because the product of abelian groups is abelian and every subgroup of an abelian group is abelian.


    Hint:

    Spoiler:


    |G/H|=[G:H]=4=2^2. Aren't all groups of order p^2 abelian? Why?
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  3. #3
    MHF Contributor Bruno J.'s Avatar
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    Quotients of abelian groups are always abelian!
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  4. #4
    MHF Contributor Drexel28's Avatar
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    Quote Originally Posted by Bruno J. View Post
    Quotients of abelian groups are always abelian!
    We seem to be clashing on what the OP is most likely to know today, huh? haha

    Of course this is true since the quotient group will be the image of the canonical homomorphism and the image of an abelian group under a homomorphism is always abelian.

    The OP will have to decide which he likes better I guess.
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