Symmetric groups are ambivalent

This article gives the statement, and possibly proof, of a particular group or type of group (namely, Symmetric group (?)) satisfying a particular group property (namely, Ambivalent group (?)).

Statement

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

Related facts

Stronger facts

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