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Math Help - Left and Right Cosets

  1. #1
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    Left and Right Cosets

    Let G be a group and H a subgroup of G. Let a, b in G. Prove the following:

    If aH = Ha and bH = Hb, then (ab)H = H(ab).

    Can someone show this please? Thanks a lot in advance!
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    Re: Left and Right Cosets

    Quote Originally Posted by jzellt View Post
    Let G be a group and H a subgroup of G. Let a, b in G. Prove the following:

    If aH = Ha and bH = Hb, then (ab)H = H(ab).

    Can someone show this please? Thanks a lot in advance!
    So we want to show that (ab)H \subset H(ab)

    let k \in (ab)H

    Then k=(ab)h for some h \in H

    By the relation bH=Hb we know that there exists an h_1 \in H such that bh=h_1b

    and we know that by aH=Ha there exists an h_2 \in H such that ah_1=h_2a

    Putting all of these together we get

    k=(ab)h=a(bh)=a(h_1b)=(ah_1)b=h_2(ab) \in H(ab)

    Now you just need to show or justify the other direction (it is exactly the same.)
    Thanks from jzellt
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