symmetric group

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 permutationsinduces a group structure on $S(X)$.We call this group the symmetric groupand it is often denoted $\mathrm{Sym}(X)$.

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.