TITLE: If $a$ commutes with $b$, does $a^{-1}$ commute with $b$?
QUESTION [12 upvotes]: $a,b$ are group elements. If $a$ commutes with $b$, does $a^{-1}$ commute with $b$?
And how can I prove this?

REPLY [2 votes]: @parsec Now you have seen all the answers you probably can prove that any power $a^n$ commutes with $b$, in other words, the group generated by your $a$ and $b$, $<a,b>$,  is abelian.