Abstract Algebra Theory and Applications - Computer Science ...

12.1 GROUPS ACTING ON SETS 205If G acts on a set X and x, y ∈ X, then x is said to be G-equivalent toy if there exists a g ∈ G such that gx = y. We write x ∼ G y or x ∼ y if twoelements are G-equivalent.Proposition 12.1 Let X be a G-set. Then G-equivalence is an equivalencerelation on X.Proof. The relation ∼ is reflexive since ex = x. Suppose that x ∼ y forx, y ∈ X. Then there exists a g such that gx = y. In this case g −1 y = x;hence, y ∼ x. To show that the relation is transitive, suppose that x ∼ yand y ∼ z. Then there must exist group elements g and h such that gx = yand hy = z. So z = hy = (hg)x, and x is equivalent to z.□If X is a G-set, then each partition of X associated with G-equivalenceis called an orbit of X under G. We will denote the orbit that contains anelement x of X by O x .Example 6. Let G be the permutation group defined byG = {(1), (123), (132), (45), (123)(45), (132)(45)}and X = {1, 2, 3, 4, 5}. Then X is a G-set. The orbits are O 1 = O 2 = O 3 ={1, 2, 3} and O 4 = O 5 = {4, 5}. Now suppose that G is a group acting on a set X and let g be an elementof G. The fixed point set of g in X, denoted by X g , is the set of all x ∈ Xsuch that gx = x. We can also study the group elements g that fix a givenx ∈ X. This set is more than a subset of G, it is a subgroup. This subgroupis called the stabilizer subgroup or isotropy subgroup of x. We willdenote the stabilizer subgroup of x by G x .Remark. It is important to remember that X g ⊂ X and G x ⊂ G.Example 7. Let X = {1, 2, 3, 4, 5, 6} and suppose that G is the permutationgroup given by the permutations{(1), (12)(3456), (35)(46), (12)(3654)}.Then the fixed point sets of X under the action of G areX (1) = X,X (35)(46) = {1, 2},X (12)(3456) = X (12)(3654) = ∅,