# What is the definition of the least common multiple of elements of a group rather than their orders?

In an answer to my question here Relationship between order of $$(x,y)$$ in $$G \times G'$$ and order of disjoint permutations, which both involve LCMs?, it is said that

Let $$G$$ be a group, $$a\in G$$ of order $$r$$ and $$b\in G$$ of order $$s$$.

If $$ab=ba$$, then the order of $$ab$$ is $$\operatorname{lcm}(a,b)$$.

I submitted an edit to change this to $$\operatorname{lcm}(r,s)$$, but I was denied. Here is the denial: https://math.stackexchange.com/review/suggested-edits/1085060 Later this was edited by José Carlos Santos into the current revision. Here is the edit into the current revision: https://math.stackexchange.com/revisions/2954951/2

The one who is answered is Scientifica. I asked him/her what $$\operatorname{lcm}(a,b)$$. The reply was

@JackBauer It means "the least common multiple of a and b". You used it yourself in your question ;)

1. What is the definition of the least common multiple of elements of a group? I could not find this online. I thought LCM was for integers under the addition operation and under the usual topology on $$\mathbb Z$$ (and under the usual everything in every branch of mathematics).

2. Where did I use the concept of least common multiple of elements of a group in my question?

• Scientifica has now clarified that your suggested edit was correct (the rejection must have been a mistake). There is no such thing as least common multiple in a group. – Arnaud D. Oct 14 '18 at 12:28