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?

I recommend an approach by contrapositive. If $aba=e,$ then $a(ba)=e$ (so $ba=a^{-1}$) and $(ab)a=e$ (so $ab=a^{-1}$), and so....
