Prove that if x = x^-1 for all x in the group G, then G is abelian. I'm getting nowhere with this after several attempts. Can someone offer insight?
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Let $\displaystyle a,b\in G$. We need to show that $\displaystyle ab=ba$. Since $\displaystyle ab = (ab)^{-1}$ we have $\displaystyle ab=(ab)^{-1}= b^{-1}a^{-1}= ba$
Alternatively, $\displaystyle a=a^{-1}\implies a^2=e$. So $\displaystyle ab=a\left(ab\right)^2b=\left(a^2\right)ba\left(b^2 \right)=ba$
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