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Math Help - H and K are subgroups of G, K is also normal in G. I want a counterexample...

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    H and K are subgroups of G, K is also normal in G. I want a counterexample...

    hello everyone. Can any one help me on this:

    H and K are subgroups of G, K is also normal in G.
    I want to find a counterexample that HK is not normal in G.

    Thanks!!
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  2. #2
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    Quote Originally Posted by xxie View Post
    hello everyone. Can any one help me on this:

    H and K are subgroups of G, K is also normal in G.
    I want to find a counterexample that HK is not normal in G.

    Thanks!!
    G=S_4, \ H=\{(1) , (1 \ 2) \}, and K=V, the Klein four group.
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    what is K=V????
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    Quote Originally Posted by xxie View Post

    what is K=V????
    i mentioned that, didn't i? it's the Klein four group: V=\{(1), (1 \ 2)(3 \ 4), (1 \ 3)(2 \ 4), (1 \ 4)(2 \ 3) \}.
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    oh~thanks! i didn't know what klein group is!haha
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    Senior Member TheAbstractionist's Avatar
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    Quote Originally Posted by xxie View Post
    hello everyone. Can any one help me on this:

    H and K are subgroups of G, K is also normal in G.
    I want to find a counterexample that HK is not normal in G.

    Thanks!!
    Hi xxie.

    Here is a much simpler example. Take G=S_3, H=\{1,(12)\}, K=\{1\}.
    Last edited by TheAbstractionist; May 7th 2009 at 03:46 AM.
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    Quote Originally Posted by TheAbstractionist View Post
    Hi xxie.

    Here is a much simpler example. Take G=S_3, H=\{1,(12)\}, K=\{1\}.
    allowing K = (1) will make the problem trivial and non-interesting, isn't it?
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    Quote Originally Posted by NonCommAlg View Post
    allowing K = (1) will make the problem trivial and non-interesting, isn't it?
    Hi NonCommAlg.

    As xxie didn’t state any condition on what type of normal subgroup K should be, my example was a fair one.

    But, if you like, how about

    G=D_6=\left<\rho,\sigma:\rho^6=\sigma^2=1,\, \rho\sigma=\sigma\rho^{-1}\right>

    H=\{1,\sigma\}

    K=\{1,\rho^3\}=Z\left(D_6\right)

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