1) The flipping of a light switch can be viewed as a group: the set would have two elements, and those elements would be the two actions “flip” and “do nothing”. The abstract group multiplication is composition of actions, so that the identity element is “do nothing” since any action composed with “do nothing” is the same as the original action. Moreover, the inverse of “do nothing” is “do nothing”, and the inverse of “flip” is “flip”, since two flips takes you back to the action that is equivalent to “do nothing”. Clearly, this multiplication is associative.
2) Clocks. These are the very reason why we call the type of addition in the solution to lesson 22 “clock arithmetic”. You could consider a 24 hour clock or a 12 hour clock, and you could consider minutes as well as hours, or just hours. If you just consider hours on a 24 hour clock, then the 24 elements of the set are and addition of hours past 23 just takes you “back around” the clock, as we normally do when telling time. I’ll leave it to you to fill in the details.
3) If driving in reverse could lower your total mileage, then the set of “adjustments of one’s total mileage” would be a group. Again, I’ll leave it to you to fill in the details, but we have “not going anywhere” as the identity element, and composition of routes as multiplication. The inverse of an element is taking that route in reverse, and so on. So I guess the set would be “the set of all drives from a given starting point”. This is clearly infinite, but that’s fine. To invert anything, I just drive that same thing in reverse. Welp, it looks like there aren’t too many more details left…
But I’m more interested in what you thought of!
On to Lesson 24