I recently learned to solve Rubik's Cubes. Along the way I learned that if you repeat any set of moves long enough, the cube will go back the way it was eventually.
I've been wondering for a couple days now if there's some formal mathematical name or concept for talking about the number of times you have to compose an operation in order for it to be idempotent.
In rubiks notation ... R rotates the right slice clockwise. Do that another three times and you can't tell you did it. Doing (R R R R) any integer number of times (gte 0) gives the same cube you started with. So my idea is to say the idempotence number of R is 4. (R D) would have to be repeated 146 times to get the same cube -- I read that somewhere, and I could be remembering it wrong. But say it's right... idempotence number 146 for the operation (R D).
I can't find anything discussions about something like this, but without knowing the name of it, I may just not know how to look it up. On the other hand, maybe it's not a worthwhile notion to think about.