# Number of equivalence classes under real conjugacy

This article defines an arithmetic function on groups

View other such arithmetic functions

## Definition

The **number of equivalence classes under real conjugacy** for a group is defined as the number of equivalence classes under the following equivalence relation: two elements are equivalent if they are either in the same conjugacy class or if the inverse of one element is in the conjugacy class of the other.

## Facts

### Ways of measuring this for a finite group

- The number of equivalence classes under real conjugacy equals the number of irreducible representations over the field of real numbers. Note: This is
*not*the number of absolutely irreducible representations over the reals.`Further information: number of irreducible representations over reals equals number of equivalence classes under real conjugacy`

### Relation with number of conjugacy classes

We have the relation with the number of conjugacy classes:

Number of equivalence classes under real conjugacy Number of conjugacy classes 2 * (number of equivalence classes under real conjugacy) - 1

For a finite group, the first inequality becomes an equality if and only if the group is an ambivalent group. The second inequality becomes an equality if and only if the group is an odd-order group.

### Relation with number of conjugacy classes and number of conjugacy classes of real elements

We have:

Number of equivalence classes under real conjugacy = (number of conjugacy classes + number of conjugacy classes of real elements)/2 = number of conjugacy classes of real elements + (1/2)(number of conjugacy classes - number of conjugacy classes of real elements)

## Relation with other arithmetic functions

### Numbers at least as big

Arithmetic function | Reason it's bigger (subset or quotient)? | Case of equality (when both numbers are finite) |
---|---|---|

number of conjugacy classes | quotient (under identification of conjugacy class with inverse conjugacy class) | ambivalent group (every element is conjugate to its inverse) |

### Numbers at most as big

Arithmetic function | Reason it's smaller (subset or quotient)? | Case of equality |
---|---|---|

number of conjugacy classes of real elements | subset | ambivalent group (every element is conjugate to its inverse) |

number of equivalence classes under rational conjuacy | quotient (under equivalence relation identifying two elements that generate the same cyclic subgroup) | ? |

number of conjugacy classes of rational elements | subset or subquotient |