# Math Help - Proving a group to be abelian

1. ## Proving a group to be abelian

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?

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

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