Number of conjugacy classes

Definition
The number of conjugacy classes in a group is defined as the number of defining ingredient::conjugacy classes, viz the number of equivalence classes under the equivalence relation of being conjugate.

This number is also sometimes termed the class number of the group.

Related group properties

 * A group in which every conjugacy class is finite is termed an FC-group. In particular, a FC-group is finite if and only if it has finitely many conjugacy classes.
 * A group with two conjugacy classes is a nontrivial group with exactly one conjugacy class of non-identity elements. Note that the only such finite group is cyclic group:Z2. However, there are many infinite groups with this property.

Ways of measuring this number for a finite group

 * The number of conjugacy classes in a finite group equals the number of equivalence classes of irreducible representations.
 * The number of conjugacy classes is the product of the order of the group and the commuting fraction of the group, which is the probability that two elements commute. This follows from the orbit-counting theorem.

Lower bounds on the number of conjugacy classes

 * The number of conjugacy classes in a group is at least as much as the number of elements in the center.
 * The number of conjugacy classes in a group is at least as much as the number of elements in the abelianization.
 * There are finitely many finite groups with bounded number of conjugacy classes. Thus, the number of conjugacy classes puts an upper bound on the order of a finite group. Conversely, the order of a group puts a lower bound on the number of conjugacy classes. On the other hand, there do exist infinite groups with only finitely many conjugacy classes.

Upper bounds on the number of conjugacy classes

 * The number of conjugacy classes is at most as much as the number of elements in the group. Equality (for finite groups) holds if and only if the group is abelian.
 * Commuting fraction more than five-eighths implies abelian: In particular, this means that for a finite non-abelian group, the number of conjugacy classes is bounded by $$5/8$$ times the order of the group.