Group theory is all about symmetries. Can this be seen from the axioms defining a group? Or equivalently can the group axioms be motivated from this point of view? Of course one can look at several examples and check that the group axioms are fulfilled, nevertheless this doesn't make clear why the axioms have to be precisly like they are. Any ideas would be much appreciated.
The most general interpretation of symmetry is a bijection of a set onto itself. The group axioms model sets of such bijections under composition. Cayley's theorem shows this interpretation can be made concrete.