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Math Help - normal subgroup

  1. #1
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    normal subgroup

    Let H be a group and K a subgroup of H with (H:K)=n< ∞.
    Prove: if (H:K)=2, then K is a normal subgroup of H
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    Quote Originally Posted by apple2009 View Post
    Let H be a group and K a subgroup of H with (H:K)=n< ∞.
    Prove: if (H:K)=2, then K is a normal subgroup of H
    The subgroup K partitions H into left cosets, one of which is K itself. If (H:K)=2, then there is only one other left coset, namely everything in H that isn't in K. The same applies to right cosets. Thus when (H:K)=2, each left coset is also a right coset. But that is one way of expressing the condition for K to be normal.
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