Conjugacy Class - Conjugacy Class Equation

Conjugacy Class Equation

If G is a finite group, then for any group element a, the elements in the conjugacy class of a are in one-to-one correspondence with cosets of the centralizer C_G(a). This can be seen by observing that any two elements b and c belonging to the same coset (and hence, b=cz for some z in the centralizer C_G(a)) give rise to the same element when conjugating a: bab−1=cza(cz)−1=czaz−1c−1=czz−1ac−1=cac−1.

Thus the number of elements in the conjugacy class of a is the index of the centralizer C_G(a) in G. Thus the size of each conjugacy class is a divisor of the order of the group.

Furthermore, if we choose a single representative element x_i from every conjugacy class, we infer from the disjointedness of the conjugacy classes that |G| = ∑_i, where C_G(x_i) is the centralizer of the element x_i. Observing that each element of the center Z(G) forms a conjugacy class containing just itself gives rise to the following important class equation:

|G| = |Z(G)| + ∑_i

where the second 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.