Symmetric groups are ambivalent

Statement
The symmetric group on any set is an ambivalent group: every element is conjugate to its inverse.

Stronger facts

 * Weaker than::Symmetric groups are rational
 * Weaker than::Symmetric groups are rational-representation
 * Weaker than::Symmetric groups are strongly ambivalent

Related facts about alternating groups

 * Classification of ambivalent alternating groups
 * Classification of alternating groups having a class-inverting automorphism
 * Finitary alternating group on infinite set is ambivalent

General information

 * Linear representation theory of symmetric groups