Proving a group to be abelian

**mjlaz** · March 1st 2010, 06:52 AM

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?

**Dinkydoe** · March 1st 2010, 07:11 AM

Let $a,b\in G$ . We need to show that $ab=ba$ .
Since $ab = (ab)^{-1}$ we have $ab=(ab)^{-1}= b^{-1}a^{-1}= ba$

**Drexel28** · March 1st 2010, 01:15 PM

Alternatively, $a=a^{-1}\implies a^2=e$ . So $ab=a\left(ab\right)^2b=\left(a^2\right)ba\left(b^2 \right)=ba$

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