TITLE: Proof that if $a,b \in G$ and $a^4b = ba$ and $a^3 = e$ then $ab = ba$
QUESTION [7 upvotes]: I tried to prove one of the examples in my Abstract Algebra book that stated: 

Prove that if $a,b \in G$ and $a^4b = ba$ and $a^3 = e$ then $ab = ba$

I went about just saying that $a^4b = ba \iff a^3(ab) = ba \iff e(ab) = ba$ and the result follows.
However, the book takes a longer route and proves it this way:

$a^4b = ba \implies b = a^6b = a^2ba \implies ab = a^3ba = ba$

Are both of our proofs valid and equivalent or am I missing something?
Thanks

REPLY [2 votes]: Another way of going around this proof: $$ab = aeb = aa^3b = a^4b = ba$$