I'm currently working on a problem where I have to find the order of xy in an abelian (not necessarily finite) group $G$, where I know that $o(x)=n$ and $o(y)=m$ are finite, and that $\gcd(n, m) = 1$. The proof I currently have is the following:
Because the $\gcd$ of $n$ and $m$ is equal to $1$, we know that $\text{lcm}(n, m) = nm$. From this, we know that there does not exists $a \leq nm$ such that $n | a$ and $m | a$. Because of this, $nm$ is the first value in $\mathbb{N}$ for which $x^{nm} = e$ and $y^{nm} = e$, so $(xy)^{nm} = x^{nm} \cdot y^{nm} = e^m \cdot e^n = e$. By the definition of the order, nm has to be the order of $xy$.
I know that my proof is wrong somewhere, but I can't really seem to find the problem, could anyone maybe help me? Thanks in advance!
