## Prove a Group is Abelian if $(ab)^3=a^3b^3$ and No Elements of Order $3$

## Problem 402

Let $G$ be a group. Suppose that we have

\[(ab)^3=a^3b^3\]
for any elements $a, b$ in $G$. Also suppose that $G$ has no elements of order $3$.

Then prove that $G$ is an abelian group.

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