If two group elements satisfy $aba = bab$, then $a^5=1$ iff $b^5=1$

Question

Let $(G,*)$ be a group where $a$ and $b$ are two elements in the group such that $aba=bab$. Prove that $a^5=e$ if and only if $b^5=e$.

I tried to manipulate the initial relationship by multiplying with the inverses to the left and to the right in order to isolate one term but I couldn't get anything useful. What is the correct approach?

Answer 1

Write $h=ab$, then $aba=bab$ means $$hah^{-1}=b.$$ We know that conjugated elements have the same order, so $a^p=e$ if and only $b^p=e$ for every number $p$.

References:

How do two conjugate elements of a group have the same order?

Prove that any conjugate of $a$ has the same order as $a$.