[mrkandel]

My professor explained that Galois originally thought about the subgroups of S_n, i.e. the set of all the permutations of n objects.

Thinking about the permutations of n objects is rather natural, especially when thinking about roots of polynomials, as when you look at roots of some polynomial you will see that some permutations of the roots are legal and some are not. And then when you investigate that you start to notice there are sets of permutation that can operate independently of the other permutation, and those are the subgroups. The concept of a "group" as an abstract term in itself was not there. Later on it was codified.