Cases where string concatenation is commutative
Simple solution: $a\cdot aa = aa \cdot a$?

In general there is a characterization due to Lyndon and Schützenberger. 
For nonempty words $x,y\in \Sigma^+$ the following are equivalent:


$xy=yx$
there exists $z\in \Sigma^+$ such that $x=z^k$ and $y=z^\ell$ for some $k,\ell &gt;0$.
there exist $i,j&gt;0$ such that $x^i = y^j$.


This means that $x$ and $y$ commute ($xy=yx$) iff they are powers of the same string, like $bbabba\cdot bbabbabba = bbabbabba \cdot bbabba$. So, basically the examples will not get more complicated than the proposal $a\cdot aa = aa \cdot a$ above.