Symmetric groups are strongly ambivalent

For a finite set
For any natural number $$n$$, the fact about::symmetric group $$S_n$$ of degree $$n$$ is a fact about::strongly ambivalent group. In other words, every element of $$S_n$$ is a fact about::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