# Ambivalent group

This article defines a group property: a property that can be evaluated to true/false for any given group, invariant under isomorphism

View a complete list of group propertiesVIEW RELATED: Group property implications | Group property non-implications |Group metaproperty satisfactions | Group metaproperty dissatisfactions | Group property satisfactions | Group property dissatisfactions

This term is related to: linear representation theory

View other terms related to linear representation theory | View facts related to linear representation theory

## Definition

### Symbol-free definition

A group is said to be **ambivalent** if every element in it is conjugate to its inverse.

For a finite group,this is equivalent to saying that every character of the group over complex numbers, is real-valued.

An element in a group that is conjugate to its inverse is termed a real element. Thus, a group is ambivalent if and only if all its elements are real elements.

### Definition with symbols

A group is said to be **ambivalent** if, for any , there exists such that .

For a finite group , this is equivalent to saying that any representation with character , for all .

## Examples

### Extreme examples

- The trivial group is ambivalent.

### Important families of groups

- Symmetric groups are ambivalent: All the symmetric groups are ambivalent.
- Classification of ambivalent alternating groups: The alternating group of degree is ambivalent only if .
- Special linear group of degree two is ambivalent iff -1 is a square
- Dihedral groups are ambivalent
- Generalized dihedral groups are ambivalent

### Groups satisfying the property

Here are some basic/important groups satisfying the property:

GAP ID | |
---|---|

Cyclic group:Z2 | 2 (1) |

Klein four-group | 4 (2) |

Symmetric group:S3 | 6 (1) |

Here are some relatively less basic/important groups satisfying the property:

GAP ID | |
---|---|

Alternating group:A5 | 60 (5) |

Alternating group:A6 | 360 (118) |

Dihedral group:D8 | 8 (3) |

Quaternion group | 8 (4) |

Symmetric group:S4 | 24 (12) |

Symmetric group:S6 | 720 (763) |

Here are some even more complicated/less basic groups satisfying the property:

GAP ID | |
---|---|

Dihedral group:D16 | 16 (7) |

Direct product of D8 and Z2 | 16 (11) |

Mathieu group:M9 | 72 (41) |

### Groups dissatisfying the property

Here are some basic/important groups that do not satisfy the property:

Here are some relatively less basic/important groups that do not satisfy the property:

GAP ID | |
---|---|

Alternating group:A4 | 12 (3) |

Here are some even more complicated/less basic groups that do not satisfy the property:

GAP ID | |
---|---|

Alternating group:A7 | |

M16 | 16 (6) |

Semidihedral group:SD16 | 16 (8) |

## Metaproperties

## Relation with other properties

### Stronger properties

Property | Meaning | Proof of implication | Proof of strictness (reverse implication failure) | Intermediate notions |
---|---|---|---|---|

rational-representation group | ||||

rational group | any two elements generating the same cyclic subgroup are conjugate. | rational implies ambivalent | ambivalent not implies rational | |FULL LIST, MORE INFO |

strongly ambivalent group | every non-identity element is either an involution or a product of two involutions | follows from strongly real implies real | ambivalent not implies strongly ambivalent | |FULL LIST, MORE INFO |

group with two conjugacy classes | there are two conjugacy classes of elements. | Rational group|FULL LIST, MORE INFO |

### Weaker properties

### Conjunction with other properties

Property | Meaning | Result of conjunction | Proof |
---|---|---|---|

abelian group | any two elements commute; or equivalently, any two conjugate elements are equal. | elementary abelian 2-group | ambivalent and abelian iff elementary abelian 2-group |

nilpotent group | admits a central series. | must be a nilpotent ambivalent 2-group | nilpotent and ambivalent implies 2-group |

odd-order group | finite group and its order is odd. | trivial group | odd-order and ambivalent implies trivial |