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Math Help - Normal subgroups

  1. #1
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    Normal subgroups

    Show that if H and K are normal subgroups of a group G such that H∩K = {e}, then hk = kh for all h ∈ H and for all k ∈ K.

    This was in our study guide for the final and a couple friends of mine and I couldn't figure it out.
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  2. #2
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    Quote Originally Posted by DanielThrice View Post
    Show that if H and K are normal subgroups of a group G such that H∩K = {e}, then hk = kh for all h ∈ H and for all k ∈ K.

    This was in our study guide for the final and a couple friends of mine and I couldn't figure it out.
    Hint: hkh^{-1}k^{-1} = h(kh^{-1}k^{-1}) = (hkh^{-1})k^{-1} \in H\cap K.
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  3. #3
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    So Essentially what we want to show is that hkh-1k-1=e.
    But H∩K={e}, so we just have to show that hkh-1k-1∈ H and hkh-1k-1∈ K, but how do we do this?
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  4. #4
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    Quote Originally Posted by DanielThrice View Post
    So we just have to show that hkh-1k-1∈ H and hkh-1k-1∈ K, but how do we do this?
    hkh^{-1}k^{-1} = h(kh^{-1}k^{-1}); and H is normal, which tells you that kh^{-1}k^{-1}\in H. So hkh^{-1}k^{-1} is the product of two elements of H and is therefore in H. Bracketing it the other way shows that the same element is also in K.
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  5. #5
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    'm sorry I've tried a couple ways to solve it for K also but it's not working for this same set up
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  6. #6
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    Quote Originally Posted by DanielThrice View Post
    'm sorry I've tried a couple ways to solve it for K also but it's not working for this same set up
    hkh^{-1}k^{-1} = (hkh^{-1})k^{-1}.
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