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  1. #1
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    Hi people,

    (G;T) is a group , "e" is it's identity element.

    I must show that if \ \ (\forall x \in G) x^2=e, (G;T) is an abelian group.

    Can you help me please???
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  2. #2
    MHF Contributor Swlabr's Avatar
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    Quote Originally Posted by bhitroofen01 View Post
    Hi people,

    (G;T) is a group , "e" is it's identity element.

    I must show that if \ \ (\forall x \in G) x^2=e, (G;T) is an abelian group.

    Can you help me please???
    Hint: ab=ba \Leftrightarrow aba^{-1}b^{-1} = e, and note that (ab)^2=e.
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  3. #3
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    Quote Originally Posted by bhitroofen01 View Post
    Hi people,

    (G;T) is a group (e is it's identity element),

    I must show that : \ (\forall x \in G)\ \ x^2=e \Longrightarrow (G;T) is commutative.

    Can you help me please???

    Hint: \forall\,x,y\in G\,,\,\,e=(xy)^2=xyxy ...

    Tonio
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  4. #4
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    \forall\,x,y\in G\,,\,\,e=(xy)^2=xyxy=yxyx=(yx)^2
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  5. #5
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    Quote Originally Posted by bhitroofen01 View Post
    \forall\,x,y\in G\,,\,\,e=(xy)^2=xyxy=yxyx=(yx)^2

    How do you justify the 3rd equality above?? Think: x^2=e\Longleftrightarrow x=x^{-1}

    Tonio
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  6. #6
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    \forall\,x,y\in G\,,\,\,e=(xy)^2=xyxy=xyy^{-1}x^{-1}
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  7. #7
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    But why aba^{-1}b^{-1} = e \Leftrightarrow ab=ba
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  8. #8
    MHF Contributor Drexel28's Avatar
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    Quote Originally Posted by bhitroofen01 View Post
    But why aba^{-1}b^{-1} = e \Leftrightarrow ab=ba
    aba^{-1}b^{-1}\implies ab=e\left(a^{-1}b^{-1}\right)^{-1}=e\left(ba\right)=ba
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