10
$\begingroup$

Let $G$ be an abelian group. Let $a, b \in G $ and let their order be $m$ and $n$ be respectively. Is it always true that order of $ab$ is $lcm(m,n)$?

What if $m$ and $n$ are coprime to each other?

  • 19
    Certainly the order of $ab$ need not be the lcm. As a simple example, $a$ and $a^{-1}$ have the same order, but $a^{-1}a$ has very small order.2011-09-24

2 Answers 2