If $n$ is odd, the symmmetry group of a regular $n$ gon (also called a dihedral group) has the property that given two commuting elements $a,b$ it follows that $a,b$ lie in a cyclic subgroup. [This includes an element commuting with its inverse, since $a^{-1}$ is a positive power of $a$ in a finite group.] Cyclic groups also (trivially) have this property. I also found that the eight element quaternion group has this property.
I'm wondering if there are other finite groups with this property. Thanks for any feedback/references.