Math and science::Algebra::Aluffi

Groups. Order for commutative product.

If $g h = h g$ , then $| g h |$ divides $lcm (| g |, | h |)$ .

Let $N = lcm (| g |, | h |)$ .

\begin{aligned} (g h)^{N} & = g^{N} h^{N} & (by commutitivity) \\ = e e & (as both | g | and | h | divide N) \\ = e \end{aligned}

By Lemma 1.10 in Aluffi, it follows that $| g h |$ divides $lcm (| g |, | h |)$ .

The proposition states "divides" and doesn't say "equal". Some examples where equality does not hold:

If $h = g^{- 1}$ , then $| g h | = 1$ , regardless of $| g |$ .
If $g = [1]_{10}$ and $h = [4]_{10}$ then $g h g h = [0]_{10}$ , so $| g h | = 2$ , but $| g | = 10$ and $| h | = 5$ .

If $g$ and $h$ commute and $\gcd (| g |, | h |) = 1$ , then $| g h | = | g | | h |$ .

Proof:

Is there a nice visualization for this?