Let $a,b\in G$. If $a^{4}b=ba$ and $a^{3}=e$, then prove that $ab=ba$.

Question

Can someone tell me whether my solution is incorrect or too short?

Let $a,b\in G$. If $a^{4}b=ba$ and $a^{3}=e$, then prove that $ab=ba$.

$ab =aeb=aa^{3}b=a^{4}b=ba$ is all I did so far. Is it too short or incorrect logically?

Answer 1

Your proof is good and correct, but the reverse order seems clearer to me: $$ ba = a^{4}b = a^3ab = eab = ab $$

Answer 2

That's correct.

An easier way to think about it once you are used to actions by conjugation is that $a=ea=a^3a=a^4=bab^{-1}$ and thus $ab=ba$. Your proof is just as good, though.