# Number of conjugacy classes of real elements

This article defines an arithmetic function on groups

View other such arithmetic functions

## Contents

## Definition

The **number of conjugacy classes of real elements** in a group, also sometimes called the **number of real conjugacy classes**, is the number of conjugacy classes in the group whose elements are real elements, i.e., conjugate to their inverses.

Note that if two elements are in the same conjugacy class, then one is a real element if and only if the other is, so this definition makes sense.

The term *number of real conjugacy classes* may sometimes (though not usually) also be used for the somewhat different notion of number of equivalence classes under real conjugacy.

## Relation with other arithmetic functions

### Bigger arithmetic functions

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

number of conjugacy classes | subset | ambivalent group |

number of equivalence classes under real conjugacy | subset | ambivalent group |

Note that the number of conjugacy classes of real elements, number of equivalence classes under real conjugacy, and number of conjugacy classes form an arithmetic progression, i.e., the number of equivalence classes under real conjugacy is the arithmetic mean of the number of conjugacy classes of real elements and the number of conjugacy classes. This is because it counts each *pair* of non-real conjugacy classes (a class and its inverse) as one.

### Smaller arithmetic functions

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

number of conjugacy classes of rational elements | subset |