Q:

Order of product of abelian group

This a question from Algebra by Artin.
Let a,b be elements of an abelian group of orders m,n respectively. What can you say about the order of their product ab?
Here is my attempt:
$a^m=1=b^n$
$a^mb^n=1$
$b^n=a^{-m}$
$b^{n-1}=a^{-(m-1)}$
(Not sure if i can assume above step)
So:
$aa^{m-1}bb^{n-1}=aa^{m-1}a^{-(m-1)}b=1$
$ab=1$
So the order of the product $ab$ is $1$

A:

Suppose
$$ord(a)=m\;,\;\;ord(b)=n\;,\;\;\text{gcd}\,(m,n)=d\implies xm+yn=d\;,\;\;x,y\in\Bbb Z\;,\;\;m,n,d\in\Bbb N$$
Now, define $\;c:=ab\;$ :
$$c^{mn/d}\stackrel{\text{why?}}=a^{mn/d}\;b^{mn/d}=\left(a^m\right)^{n/d}\left(b^n\right)^{m/d}=1\cdot1=1$$
(Observe that $\;\frac nd\,,\,\,\frac md\in\Bbb Z\;$)
But of course
$$\frac{mn}d=\text{lcm}\,(m,n)\;\;,\;\;\;\text{and we're done}$$

