Group Permutations Proof If $a^6 = e$ and $ab = ba^2$, show that $a^3 = e$ and $aba = b$ Workings: Proof: $aba = (ba^2)a$ $aba = ba^3$ $aba = be$ $aba = b$ I'm not sure how to show that $a^3 = e$ though. Any help will be appreciated.

$ab=ba^2 \Rightarrow b^{-1}ab=a^2 \Rightarrow b^{-1}a^3b=a^6=e \Rightarrow a^3=e.$