Dihedral Group of Order 6 - Conjugacy Classes

Conjugacy Classes

We can easily distinguish three kinds of permutations of the three blocks, called conjugacy classes of the group:

• no change, a group element of order 1
• interchanging two blocks: (RG), (RB), (GB), three group elements of order 2
• a cyclic permutation of all three blocks (RGB), (RBG), two group elements of order 3

For example (RG) and (RB) are both of the form (x y); a permutation of the letters R, G, and B (namely (GB)) changes the notation (RG) into (RB). Therefore, if we apply (GB), then (RB), and then the inverse of (GB), which is also (GB), the resulting permutation is (RG).

Note that conjugate group elements always have the same order, but for groups in general group elements that have the same order need not be conjugate.