Number of equivalence classes under real conjugacy

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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

Relation with number of conjugacy classes

We have the relation with the number of conjugacy classes:

Number of equivalence classes under real conjugacy \le Number of conjugacy classes \le 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