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Math Help - Proving a group is Abelian...

  1. #1
    Junior Member ginafara's Avatar
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    Proving a group is Abelian...

    Can someone help me with this...TIA

    Let G be a group which satisfies the following property:
    Whenever x,y,z are elements of G such that xy=yz, then x=z.
    Prove that G must be Abelian.


    I don't even know how to start. TIA...
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  2. #2
    is up to his old tricks again! Jhevon's Avatar
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    Quote Originally Posted by ginafara View Post
    Can someone help me with this...TIA

    Let G be a group which satisfies the following property:
    Whenever x,y,z are elements of G such that xy=yz, then x=z.
    Prove that G must be Abelian.


    I don't even know how to start. TIA...
    I have NO experience with Abstract Algebra whatsoever, but for some reason i felt like taking a stab at this.

    According to Wikipedia, an Abelian group is a group (G,*) such that a*b = b*a for all a,b in G.

    now we are told that xy = yz \implies x = z, for any x,y,z in G

    this means that we can write, xy = yx or alternatively, zy = yz. Which is the exact definition of an Abelian Group.



    don't take my word for it though, wait until the experts get here
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  3. #3
    Junior Member ginafara's Avatar
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    I have no experience with A.A. either...I am strictly Applied... I will try and get it going... thanks
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  4. #4
    MHF Contributor red_dog's Avatar
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    By associativity, we have (ab)a=a(ba).
    Now, take x=ab, z=ba and y=a.
    Thus, xy=yz\Rightarrow x=z\Rightarrow ab=ba.
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