Number of conjugacy classes of real elements

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.

Bigger arithmetic functions
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.