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?

$$aba=1\implies ab=a^{-1}\;\;(1)\implies a=a^{-1}b^{-1}=(ba)^{-1}\;\;(2)$$
From (1)-(2) it follows
$$(1)\;a^{-1}=ab\implies a=(ab)^{-1}=b^{-1}a^{-1}\stackrel{(2)}=a^{-1}b^{-1}\ldots$$
