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Math Help - Abstract Algebra II: Normal Subgroups

  1. #1
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    Abstract Algebra II: Normal Subgroups

    Let H be a subgroup of a group G. Prove that
    H is a normal subgroup G if and only if Hg = gH for every g that exist G:

    (Note: Hg = {hg : h exist in Hg} and gH = {gh : h exist in Hg} are sets. So, in the proof of -->, you suppose that H is a normal subgroup of G and then prove two inclusions: Hg is a subset of gH and gH is a subset of Hg.)
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  2. #2
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    Quote Originally Posted by mathgirl1188 View Post
    Let H be a subgroup of a group G. Prove that
    H is a normal subgroup G if and only if Hg = gH for every g that exist G:

    (Note: Hg = {hg : h exist in Hg} and gH = {gh : h exist in Hg} are sets. So, in the proof of -->, you suppose that H is a normal subgroup of G and then prove two inclusions: Hg is a subset of gH and gH is a subset of Hg.)

    H\triangleleft G\Longleftrightarrow \forall g\in G\,,\,g^{-1}Hg=H\Longleftrightarrow gH=Hg

    Tonio
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