Abstract Algebra Theory and Applications - Computer Science ...

12.2 THE CLASS EQUATION 20712.2 The Class EquationLet X be a finite G-set and X G be the set of fixed points in X; that is,X G = {x ∈ X : gx = x for all g ∈ G}.Since the orbits of the action partition X,|X| = |X G | +n∑|O xi |,where x k , . . . , x n are representatives from the distinct nontrivial orbits of X.Now consider the special case in which G acts on itself by conjugation,(g, x) ↦→ gxg −1 . The center of G,i=kZ(G) = {x : xg = gx for all g ∈ G},is the set of points that are fixed by conjugation. The nontrivial orbits ofthe action are called the conjugacy classes of G. If x 1 , . . . , x k are representativesfrom each of the nontrivial conjugacy classes of G and |O x1 | =n 1 , . . . , |O xk | = n k , then|G| = |Z(G)| + n 1 + · · · + n k .The stabilizer subgroups of each of the x i ’s, C(x i ) = {g ∈ G : gx i = x i g},are called the centralizer subgroups of the x i ’s. From Theorem 12.3, weobtain the class equation:|G| = |Z(G)| + [G : C(x 1 )] + · · · + [G : C(x k )].One of the consequences of the class equation is that the order of eachconjugacy class must divide the order of |G|.Example 8. It is easy to check that the conjugacy classes in S 3 are thefollowing:{(1)}, {(123), (132)}, {(12), (13), (23)}.The class equation is 6 = 1 + 2 + 3.Example 9. The conjugacy classes for D 4 are{(1)}, {(13), (24)}, {(1432), (1234)}, {(12)(34), (14)(23), (13)(24)}.The class equation is 8 = 1 + 2 + 2 + 3.