Is a group in which $a^3 = 1$ for all $a \in G$ abelian?

Question

Suppose that $G$ is a group in which $a^3 =1$ for all $a \in G$. Is $G$ abelian?

I tried to play around with algebraic expressions. I found e.g. $(ab)^2 = b^2 a^2$ for all $a,b$. I don't know what to do.

I also can't find a counterexample.

Thanks for help

Answer 1

Nope, the simples example is problably the Heisenberg group over $\mathbb{F}_3$, which has order $27$. This is also the smallest counterexample: every group of order $p$ and $p^2$ ($p$ prime) is abelian.

Note: the Heisenberg group over $\mathbb{F}_2$ has exponent $4$.