Symmetric groups are strongly ambivalent

Statement

For a finite set

For any natural number $n$, the Symmetric group (?) $S_{n}$ of degree $n$ is a Strongly ambivalent group (?). In other words, every element of $S_{n}$ is a Strongly real element (?): it is either the identity element or is conjugate to its inverse via an element of order two.

General statement

The symmetric group on any set is a strongly ambivalent group. In other words, every element of the group is a strongly real element.

Related facts

• Symmetric groups are rational
• Symmetric groups are rational-representation
• Symmetric groups are ambivalent