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

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