Let G be a Group and $a,b \in G$. If $a$ and $b$ commute and $|a|$ and $|b|$ are finite. What can be possible values for $|ab|$?

Question

I want to find the order of $ab$, which I have tried to find as follows.
Let order of $a$ be $|a|=m$ and that of $b=|b|=n$
Let $x=\text{lcm} (m,n)$, then clearly $\exists$ integers $ s , t $ such that $x=ms$ and $x=nt$
Thus, $(ab)^x=a^x b^x=a^{ms}b^{nt}=e$
If $y\lt x$ then, $(ab)^y=a^yb^y=b^y (\text{or} a^y)$, if $y$ is some multiple of $m$, less than $x=ms$(or, if $y$ is multiple of $n$ but less than $nt$. Either way,$(ab)^y\ne e$
Therefore, $|ab|=x$=lcm$(m,n)$
Is the above reasoning correct? Can you please help me find the right solution. Thanks.