If a and b are group elements and ab ≠ ba, prove that aba ≠ identity. Q: If a and b are group elements and ab ≠ ba, prove that aba ≠ identity. 
I began by stating my theorem, then assumed ab ≠ ba. Then I tried a few inverse law manipulations, which worked in a sense, however they brought me nowhere, as I couldn't conclude my proof concretely. Suggestions? Solutions?

Try proving the contrapositive; that is, if $aba=e$ then $ab=ba$. First of all, multiply both sides of $aba=e$ on the left by $a^{-1}$ to get $ba=a^{-1}$; now multiply both sides of this on the right by $a^{-1}$ to get $b=a^{-2}$. Now, can you see why $b$ commutes with $a$?
