For any two elements g, x \in G, let g \cdot x := gxg^{-1}. This defines a
group action of G on G. The
orbits of this action are the conjugacy classes. Let \operatorname{C}_G(a) denote the
centralizer of a \in G, i.e., the
subgroup consisting of all elements b\in G such that ba = ab. Then the
stabilizer of a given element x\in G is \operatorname{C}_G(x). Moreover, the set of elements fixed by g\in G under conjugation is \operatorname{C}_G(g).
Conjugacy class equation For any element a of a group G, the elements of the conjugacy class of a are in one-to-one correspondence with
cosets of the centralizer \operatorname{C}_G(a). This can be seen by observing that any two elements b and c belong to the same coset of \operatorname{C}_G(a), meaning b = cz for some z\in\operatorname{C}_G(a), if and only if they give rise to the same element when conjugating a: bab^{-1} = cac^{-1} \Longleftrightarrow \left(c^{-1}b\right)a = a\left(c^{-1}b\right) \Longleftrightarrow z = c^{-1}b \in \operatorname{C}_G(a). This is a special case of the
orbit-stabilizer theorem, keeping in mind that conjugacy classes are orbits and centralizers are stabilizers under the action of the group on itself through conjugation. Thus if G is a
finite group, the number of elements in the conjugacy class of a is the
index \left[ G : \operatorname{C}_G(a)\right] of the centralizer \operatorname{C}_G(a) in G; hence the size of each conjugacy class divides the order of the group. Furthermore, if we choose a single representative element x_i from every conjugacy class, we infer from the disjointness of the conjugacy classes that |G| = \sum_i \left[ G : \operatorname{C}_G(x_i)\right]. Observing that each element of the center \operatorname{Z}(G) forms a conjugacy class containing just itself gives rise to the
class equation: |G| = |{\operatorname{Z}(G)}| + \sum_i \left[G : \operatorname{C}_G(x_i)\right], where the sum is over a representative element from each conjugacy class that is not in the center. Knowledge of the divisors of the group order |G| can often be used to gain information about the order of the center or of the conjugacy classes.
Example Consider a finite
p-group G (that is, a group with order p^n, where p is a
prime number and n > 0). We are going to prove that every finite p-group has a center of size greater than 1. Since the order of any conjugacy class of G must divide the order of G, it follows that each conjugacy class H_i that is not in the center also has order some power of p^{k_i}, where 0 But then the class equation requires that |G| = p^n = |{\operatorname{Z}(G)}| + \sum_i p^{k_i}. From this we see that p must divide |{\operatorname{Z}(G)}|, so |\operatorname{Z}(G)| > 1. In particular, when n = 2 we can further show that G is abelian. From the foregoing, |\operatorname{Z}(G)|, equals either p or p^2 and, if G were nonabelian, would have to equal p. Furthermore, there would have to be an element b not in \operatorname{Z}(G). Its centralizer subgroup \operatorname{C}_G(b) would, however, have to include both b and all the elements of \operatorname{Z}(G), implying |\operatorname{C}_G(b)| = p^2. This contradicts b\notin\operatorname{Z}(G). Hence G is abelian and is, in fact, isomorphic either to a
cyclic group of order p^2 or to the direct product of two cyclic groups of order p.
Average centralizer By
Burnside's lemma, the number of conjugacy classes of a finite group G is equal to \frac{1} \sum_g |C_G(g)|, the average size of the sets fixed by the elements of G acting by conjugation, that is, the average size of the centralizers of elements of G. ==Conjugacy of subgroups and general subsets== More generally, given any
subset S \subseteq G (S not necessarily a subgroup), define a subset T \subseteq G to be conjugate to S if there exists some g \in G such that T = gSg^{-1}. Let \operatorname{Cl}(S) be the set of all subsets T \subseteq G such that T is conjugate to S. Define a group action of G on the set of all
subsets of G, by writing g \cdot S := gSg^{-1}. The result for the number of elements in the conjugacy class of an element generalizes to the case of subsets using an argument analogous to the one used previously: there is a one-to-one correspondence between elements of the conjugacy class of S and cosets of \operatorname{N}(S), the
normalizer of S, since, if g, h \in G, then gSg^{-1} = hSh^{-1} if and only if g^{-1}h \in \operatorname{N}(S), in other words, if and only if g \text{ and } h are in the same
coset of \operatorname{N}(S). As a result, the index of \operatorname{N}(S) in G equals the
cardinality of \operatorname{Cl}(S): |{\operatorname{Cl}(S)}| = [G : N(S)]. By using S = \{ a \}, this formula specializes to the one given earlier for the number of elements in a conjugacy class since the normalizer of a singleton equals its centralizer. The above is particularly useful when talking about subgroups of G. The subgroups can thus be divided into conjugacy classes, with two subgroups belonging to the same class if and only if they are conjugate. Conjugate subgroups are
isomorphic, but isomorphic subgroups need not be conjugate. For example, an abelian group may have two different subgroups which are isomorphic, but they are never conjugate. ==Geometric interpretation==