Difference between permutations and a symmetric group

Question

Sorry, noob question, and it might be more of a history question. It seems to me that the symmetric group $S_n$ is the same thing as all permutations of $n$ distinct elements. Wolfram states the exactly same thing as well. So in other words, there's no difference between the two other than notation differences? Then why is it necessary to have developed two different notations/labels for the same thing?

Answer 1

A permutation can be already explained as bijection from a finite set $X$ to itself. $$ \pi : X \to X $$ Here it is a special kind of function. One can collect them in a set. If $X$ has $n$ elements, then there are $n!$ permutations of $X$.

The symmetric group emphasizes the algebraic aspects of the elements of the full sets, and sees it as special kind of group.

Answer 2

A permutation group on $n$ objects might not necessarily include all possible permutations, but if it does, it is called the symmetric group $S_n$.

Compare with graph theory, where it is convenient to have a notation for a graph on $n$ vertices where all pairs of vertices are adjacent: this is called the complete graph on $n$ vertices, or $K_n$.