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Math Help - Groups - just want to check if i'm on the right track!?

  1. #1
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    Groups - just want to check if i'm on the right track!?

    H, K sub-groups of a group G. x,y elements of G. Suppose Hx=Ky. Show that x(y^-1) is an element of H, and deduce that H = K.


    I just multiplied x^-1 to get H=Ky(x^-1), to get yx^-1 element of H, since H is a sub-group, it's inverse, y^-1x is also an element of H...

    can i say that?

    How would I show that H = K?

    Thanks!
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  2. #2
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    Quote Originally Posted by rmangan View Post
    H, K sub-groups of a group G. x,y elements of G. Suppose Hx=Ky. Show that x(y^-1) is an element of H, and deduce that H = K.
    since y \in Ky = Hx, there exists h \in H such that y=hx. hence xy^{-1}=h^{-1} \in H. thus: K=Hxy^{-1} =H.
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