Why is the number of elements in a group called "order"? This is a question that I have for a long time, Why is the number of elements in a group called "order"? I mean, the word "order" in Spanish (which is my language) has a very strong meaning in terms of "ordering", but it does not refer to quantities. What was the motivation for this?
 A: I answered this question on hsm.SE. As the question here is still open and lying in the "unanswered questions" queue, I've made this Community Wiki answer which duplicates my answer from hsm.SE.

The theory of permutations and permutation groups was the original (abstract) setting of group theory, and so the term originated there. I believe the reason for it was as follows:

The order of a permutation is the least number such that the ordering of the index set is preserved.

I looked in both Lagrange's 1771 memoir and Cauchy's 1844 memoir, which both address Lagrange's theorem on orders of subgroups. Lagrange barely uses the word order (well, "ordre" as he is writing in French), and doesn't use it in a way which influences our discussion here (the only relevant use is on p201, when he changes the orders of variables in an equation).
Cauchy's memoir is more relevant. He goes from talking about, basically, "permutations change the order of the index set" to defining the order of a permutation. He doesn't justify this name, but writing like this does suggest the reason is that it is the least number such that the ordering of the index set is preserved, as per my claim above.
