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Math Help - Subgroup

  1. #1
    MHF Contributor alexmahone's Avatar
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    Subgroup

    Show that a nonempty subset H of a group G is a subgroup of G if and only if ab^{-1} \in H for all a, b \in H.
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  2. #2
    MHF Contributor Swlabr's Avatar
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    Re: Subgroup

    Quote Originally Posted by alexmahone View Post
    Show that a nonempty subset H of a group G is a subgroup of G if and only if ab^{-1} \in H for all a, b \in H.
    Clearly this condition must hold if your set is a subgroup (as subgroups are closed under multiplication).

    If this condition holds then note that you have the identity in your set (take a=b), and that if g\in H then g^{-1} \in H (take a=1, b=g).

    So the only thing to prove is that if g, h\in H then gh\in H. However, as h^{-1}\in H, take a=g and b=h^{-1}...and your done!
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