Let $X$ be a set. Let $S(X)$ be the set of permutations of $X$ (i.e. the set of bijective functions on $X$ . Then the act of taking the composition of two permutations induces a group structure on $S(X)$ We call this group the symmetric group and it is often denoted ${\rm Sym}(X)$

When $X$ has a finite number $n$ of elements, we often refer to the symmetric group as $S_n$ and describe the elements by using cycle notation.