# Generalized dihedral groups are ambivalent

From Groupprops

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

## Contents

## Statement

Let be any abelian group and be the generalized dihedral group corresponding to . Then, is an ambivalent group: every element of is conjugate to its inverse. In particular, if is finite, all characters of are real-valued.

## Related facts

### Related facts about generalized dihedral groups

- Generalized dihedral groups are strongly ambivalent
- Classification of rational generalized dihedral groups

### Related facts about similar groups

- Dihedral groups are ambivalent
- Dicyclic groups of even degree are ambivalent
- Symmetric groups are rational
- Classification of ambivalent alternating groups

## Proof

### Proof outline

Any element in the abelian normal subgroup is conjugate to its inverse via the conjugating element. An element outside the abelian normal subgroup is an involution -- it has order two, so it is conjugate to its inverse for obvious reasons.