Section 1.7. Group Actions 65 Example 1.100. Cayley’s Theorem says that G acts on itself by translations: τg : a → ga. If a ∈ G, then the orbit O(a) = G, for if b ∈ G, then b = (ba−1)a = τba−1 (a). The stabilizer Ga of a ∈ G is {1}, for if a = τg(a) = ga, then g = 1. We say that G acts transitively on X if there is only one orbit. More generally, G acts transitively on G/H [the family of (left) cosets of a (not necessarily normal) subgroup H] by translations τg : aH → gaH. The orbit O(aH) = G/H, for if bH ∈ G/H, then τba−1 : aH → bH. The stabilizer GaH of the coset aH is aHa−1, for gaH = aH if and only if a−1ga ∈ H if and only if g ∈ aHa−1. Example 1.101. Let X = {1, 2,...,n}, let α ∈ Sn, and define the obvious action of the cyclic group G = α on X by αk · i = αk(i). If i ∈ X, then O(i) = {αk(i) : 0 ≤ k |G|}. Suppose the complete factorization of α is α = β1 · · · βt(α) and i = i1 is moved by α. If the cycle involving i1 is βj = (i1 i2 . . . ir), then the proof of Theorem 1.6 shows that ik+1 = αk(i1) for all k r. Therefore, O(i) = {i1,i2,...,ir}, where i = i1. It follows that |O(i)| = r. The stabilizer Gi of a number i is G if α fixes i however, if α moves i, then Gi depends on the size of the orbit O(i). For example, if α = (1 2 3)(4 5)(6), then G6 = G, G1 = α3 , and G4 = α2 . Example 1.102. Let X = {v0, v1, v2, v3} be the vertices of a square, and let G be the dihedral group D8 acting on X, as in Figure 1.9 (for clarity, the vertices in the figure are labeled 0, 1, 2, 3 instead of v0, v1, v2, v3). 0 1 2 3 0 1 2 3 0 1 2 3 0 2 1 3 0 1 2 3 0 1 2 3 0 1 2 3 0 1 2 3 Figure 1.9. Dihedral group D8. Thus, G = {rotations} ∪ {reflections} = {(1), (0 1 2 3), (0 2)(1 3), (0 3 2 1)} ∪ {(1 3), (0 2), (0 1)(2 3), (0 3)(1 2)}. For each vertex vi ∈ X, there is some g ∈ G with gv0 = vi therefore, O(v0) = X and D8 acts transitively.

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