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Math Help - Abstract Algebra 1

  1. #1
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    Abstract Algebra 1

    Let G be a group and let H and K be subgroups of G. Suppose there are
    a, b exist in G such that Ha = Kb. Prove that H = K.
    Hint: H and K are sets, so you must prove that H is a subset of  K and K is a subset of H.

    Any help would be very helpful. I have tried countless ways to solve this but keep getting stuck! HELP PLEASE!
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  2. #2
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    Quote Originally Posted by nconti View Post
    Let G be a group and let H and K be subgroups of G. Suppose there are
    a, b exist in G such that Ha = Kb. Prove that H = K.
    Hint: H and K are sets, so you must prove that H is a subset of  K and K is a subset of H.

    Any help would be very helpful. I have tried countless ways to solve this but keep getting stuck! HELP PLEASE!

    Denote by 1 the group's unit, then with h=1 we get 1\cdot a= kb , for some k\in K\Longrightarrow ba^{-1}=k^{-1}\in K .

    Now, let h\in H be any element, then there exists k'\in K\,\,\,s.t.\,\,\,ha=k'b\Longrightarrow h=k'(ba^{-1})=k'k^{-1}\in K\Longrightarrow h\in K\Longrightarrow H\subset K since

    the last part was true for any element in H .

    Now you prove the opposite direction in a similar way.

    Tonio
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