Conjugacy class
Conjugacy class equation

If G {\displaystyle G} is a finite group, then for any group element a, {\displaystyle a,} the elements in the conjugacy class of a {\displaystyle a} are in one-to-one correspondence with cosets of the centralizer C G (a). {\displaystyle \operatorname {C} _{G}(a).} This can be seen by observing that any two elements b {\displaystyle b} and c {\displaystyle c} belonging to the same coset (and hence, b = c z {\displaystyle b=cz} for some z {\displaystyle z} in the centralizer C G (a) {\displaystyle \operatorname {C} _{G}(a)}) give rise to the same element when conjugating a {\displaystyle a}:

That can also be seen from the orbit-stabilizer theorem, when considering the group as acting on itself through conjugation, so that orbits are conjugacy classes and stabilizer subgroups are centralizers. The converse holds as well.

Thus the number of elements in the conjugacy class of a {\displaystyle a} is the index {\displaystyle \left} of the centralizer C G (a) {\displaystyle \operatorname {C} _{G}(a)} in G {\displaystyle G}; hence the size of each conjugacy class divides the order of the group.

Furthermore, if we choose a single representative element x i {\displaystyle x_{i}} from every conjugacy class, we infer from the disjointness of the conjugacy classes that

where C G (x i) {\displaystyle \operatorname {C} _{G}(x_{i})} is the centralizer of the element x i. {\displaystyle x_{i}.} Observing that each element of the center Z (G) {\displaystyle \operatorname {Z} (G)} forms a conjugacy class containing just itself gives rise to the class equation:

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 | {\displaystyle |G|} can often be used to gain information about the order of the center or of the conjugacy classes.
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