Number of equivalence classes under real conjugacy

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.

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.

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)