Order of a product in an abelian group. Suppose $G$ is a finite abelian group and has two element $a$ and $b$, such that $\circ(a)=m$ and $\circ(b)=n$ and $lcm(m,n)\neq m,n$. Is it true that $\circ(ab)=lcm(m,n)$? Thanks in advance.

The statement is not true. In $\mathbb{Z}/30 \mathbb{Z}$, $o(3) = 10$ and $o(5) = 6$, but lcm$(6,10) = 30$ and $o(3+5) = o(8) = 15$.
