This one has been bugging me for a while - I have a number of definitions for a symmetric group, none of which I understand. Clearly, $S_n$ is a group, which I follow. Now, the wikipedia definition (most readily to hand) is:
The symmetric group on a set X is the group whose underlying set is the collection of all bijections from X to X and whose group operation is that of function composition.
Of that, I can see (clearly) we have an underlying set which combined with an operation follows set rules as per the definition of a group. I understand a bijection from X to X implies that $\forall x \in X$, $x \mapsto x$. So, just as in geometric symmetry a set is symmetrical if it becomes itself again after some map.
All well and good, but how do composite functions relate to this? Surely taking the bijective requirement into consideration the only operation permitted is one that effectively leaves the set unchanged?
So, can anyone please either explain how composite functions fit in my definition or (equally fine) provide an example of a function that works so I can try to fit the idea to the definitions I've come across.