Symmetric group on finite set

Definition
A symmetric group on finite set or symmetric group of finite degree is a symmetric group on a finite set.

See symmetric group for more general information about symmetric groups.

Small finite values
Since alternating groups are simple for degree at least five, all symmetric groups of degree at least five are not solvable. Also, all symmetric groups of degree greater than two are centerless, and among them, the one of degree six is the only one that is not complete.

Group properties
Below we discuss properties satisfied for the symmetric group of degree $$n$$.