Conjugacy Class - Properties

Properties

• The identity element is always in its own class, that is Cl(e) = {e}

• If G is abelian, then gag−1 = a for all a and g in G; so Cl(a) = {a} for all a in G; the concept is therefore not very useful in the abelian case. The failure of this thus gives us an idea in what degree the group is nonabelian.

• If two elements a and b of G belong to the same conjugacy class (i.e., if they are conjugate), then they have the same order. More generally, every statement about a can be translated into a statement about b=gag−1, because the map φ(x) = gxg−1 is an automorphism of G.

• An element a of G lies in the center Z(G) of G if and only if its conjugacy class has only one element, a itself. More generally, if C_G(a) denotes the centralizer of a in G, i.e., the subgroup consisting of all elements g such that ga = ag, then the index is equal to the number of elements in the conjugacy class of a (by the orbit-stabilizer theorem).

• If a and b are conjugate, then so are powers of them, and – thus taking kth powers gives a map on conjugacy classes, and one may speak of which conjugacy classes a given conjugacy class "powers up" into. For example, in the symmetric group, the square of an element of type (3)(2) (a 3-cycle and a 2-cycle) is an element of type (3), while the cube is an element of type (2), so the class (3)(2) powers up into the classes (3) and (2).