A symmetric group is a group consisting of all permutations of given finite set, with composition of permutations as the binary operation. Should be used with the (group-theory) tag.
The symmetric group $S_n$ is a group consisting of all permutations of a set of $n$ elements with composition as the binary operation. You could equivalently think of it as the group of all bijective functions from a set $\{1,2,\dotsc,n\}$ to itself. The symmetric group can be generated by the functions that swap adjacent pairs of elements $\{1,2,\dotsc,n\}$. This leads the a common presentation of the symmetric groups with generators $\langle \sigma_1, \sigma_2, \dotsc, \sigma_{n-1}\rangle$ and relations
- $\sigma_i^2 = 1$
- $\sigma_i\sigma_j = \sigma_j\sigma_i$ for $|i-j|>1$
- $(\sigma_i\sigma_{i+1})^3 = 1$
