Give me please an example of the isomorphic groups from the real life that is hard to notice. I mean two groups that are actually isomorphic and exist near us, but if we consider them briefly it will be almost impossible to notice an isomorphism or some kind of similiarity between them.
For example the movements of someone is a set and a composition law is applying them during some interval of time. It is associative because once I decided what movements I am going to make it does not matter for the outcome from which one I will begin. The identity element is not moving at all. And a reverse element is to move to the previous position.