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Math Help - Prove that G is abelian given that each element of G has order at most 2.

  1. #1
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    Thumbs up Prove that G is abelian given that each element of G has order at most 2.

    Prove that, if G is a group and each element of G has order at most 2, then G is abelian.

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  2. #2
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    Since a^2=e, a=a^{-1} for all a. Now look at the equation (ab)^2=e
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  3. #3
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    e is the only element of order 1.

    if g,h \in G then gh \in G
    so (gh)^2 = e
    and (gh)^2 = ghgh
    so ghgh = e
    ghghh = h
    ghg = h
    ghgg = hg
    gh = hg
    Therefore G such that all elements have order of at most 2 is Abelian.
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