Prove that a group $G$ is abelian

Question

Suppose we have a group $(G, *).$ Prove that the group is abelian if $b * a^2 = b$ where $(a, b)$ are part of the group.

Answer 1

Use the associative property to write

\begin{align*} ab &= (b^2a)(a^2b) \\ &= b(ba^2)ab \\ &= (ba)(ab) \\ &= b(a^2b) \\ &= ba. \end{align*}

Hence $G$ is Abelian.