The Conjugacy Class Equation
The elements of any group may be partitioned into conjugacy classes. Two elementsof a group are in the same conjugacy class if there existssuch that
Conjugacy is an equivalence relation:
Ifthen there existssuch thathence h sim g.
Ifandthen there existsuch thatandthen
Hence the operation of conjugation divides the group into equivalence classes. Members of the same conjugacy class share many properties – they have the same order, if they are members of a permutation group, they have the same cycle structure for example. It does not follow that all elements of the same order or type fall in the same conjugacy class. In fact, for abelian groups every elementis in a conjugacy class all by itself, sincefor allOnly non – abelian groups are of interest.
Example:has three conjugacy classes:
From this we can write down the conjugacy class equation forEach term in this equation in the number of elements in that conjugacy class. Since all the conjugacy classes are disjoint and their union is the group itself, the sum of the orders of the conjugacy classes is equal to the order of the group.
Example:has 3 conjugacy classes
The conjugacy class equation foris then 8=1+2+1+4.
Notice that the size of each conjugacy class is a divisor of the order of the group. This is a consequence of Lagrange's theorem, since the stabilizer of each conjugacy class is a subgroup, so divides the order of the group.