Example of an infinite balanced nonabelian group > A group is called a _balanced_ group provided that, for all $a,b \in G$, either $ab=ba$ or $a^2=b^2$. Example of (possibly infinite) balanced groups are abelian groups. An example of a finite balanced nonabelian group is the Quaternion group. Does it exist an infinite balanced nonabelian group?

Yes, $Q_8 \times \mathbb{Z}_2^\omega$ is balanced, where $Q_8$ is the quaternion group of order $8$ and $\mathbb{Z}_2^\omega$ is the direct sum of infinitely many copies of $\mathbb{Z}_2$.