Let {G;*} be a group such that a^2 = e for every a belonging to G. Prove that G must be a abelian group.
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Originally Posted by tigergirl Let {G;*} be a group such that a^2 = e for every a belonging to G. Prove that G must be a abelian group. $\displaystyle a.b = b.a.a^{-1}.b^{-1}.a.b$ $\displaystyle = b.a.a^{-1}.b^{-1}.(b.a).(b.a).a.b$ $\displaystyle = b.a.e.e$
$\displaystyle a^2=e$ $\displaystyle b^2=e$ $\displaystyle abab=(ab)(ab)=(ab)^2=e$ Now take this and multiply on the left by a and right by b. $\displaystyle a(abab)b=a(e)b$ $\displaystyle (aa)ba(bb)=ab$ $\displaystyle ba=ab$
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