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If $\forall x \in G$ where G is a group $x=x^{-1}$ then how is this group abelian .

This question is from Gallian's book where question provides information that G is a group and $\forall x \in G$ , $x=x^{-1}$ then we have to prove that G is abelian , how is it possible ??

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    $\begingroup$ Hint : In such a group, we have $x^2=e$ for all $x\in G$. Now, consider the products $abab$ and $aabb$ $\endgroup$
    – Peter
    Apr 12, 2018 at 20:38
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    $\begingroup$ Please search next time before posting. $\endgroup$
    – Dietrich Burde
    Apr 13, 2018 at 8:55

2 Answers 2

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We wish to show $\forall x,y \in G$, $xy = yx$. Note that $xy \in G \implies xy = (xy)^{-1} = y^{-1}x^{-1} = yx$

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We have $ab=(ab)^{-1}=b^{-1}a^{-1}=ba$

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