Conjugacy Class

A complete set of mutually conjugate Group elements. Each element in a Group belongs to exactly one class, and the identity ($I=1$) element is always in its own class. The Orders of all classes must be integral Factors of the Order of the Group. From the last two statements, a Group of Prime order has one class for each element. More generally, in an Abelian Group, each element is in a conjugacy class by itself. Two operations belong to the same class when one may be replaced by the other in a new Coordinate System which is accessible by a symmetry operation (Cotton 1990, p. 52). These sets correspond directly to the sets of equivalent operation.

Let $G$ be a Finite Group of Order $\vert G\vert$, and let $s$ be the number of conjugacy classes of $G$. If $\vert G\vert$ is Odd, then

$$\vert G\vert\equiv s\ \left({{\rm mod\ } {16}}\right)$$

(Burnside 1955, p. 295). Furthermore, if every Prime $p_i$ Dividing $\vert G\vert$ satisfies $p_i\equiv 1\ \left({{\rm mod\ } {4}}\right)$, then

$$\vert G\vert\equiv s\ \left({{\rm mod\ } {32}}\right)$$

(Burnside 1955, p. 320). Poonen (1995) showed that if every Prime $p_i$ Dividing $\vert G\vert$ satisfies $p_i\equiv 1\ \left({{\rm mod\ } {m}}\right)$ for $m\geq 2$, then

$$\vert G\vert\equiv s\ \left({{\rm mod\ } {2m^2}}\right).$$

References

Burnside, W. Theory of Groups of Finite Order, 2nd ed. New York: Dover, 1955.

Cotton, F. A. Chemical Applications of Group Theory, 3rd ed. New York: Wiley, 1990.

Poonen, B. ``Congruences Relating the Order of a Group to the Number of Conjugacy Classes.'' Amer. Math. Monthly 102, 440-442, 1995.