# Symmetric groups are ambivalent

From Groupprops

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 (?)).

## Contents

## 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