In this section, we learn a lot more about how to work with $S_n$, the group of permutations on any n objects (but, canonically, on the set $\{1,2,\dots,n\}$).
Some of this is just skill that will be usefully generally, later on. But in particular, we will see how some of what we learn can be used to define the “alternating group”, a subgroup of $S_n$.
We do not study the alternating group in much depth here. But in the next chapter we will show that for $n\ge 5$ this subgroup is a non-commutative simple group. This will have fundamental importance in Galois theory, when we prove that polynomials of degree 5 “are not solvable” in some sense that will be defined later.