Abstract Algebra Theory and Applications - Computer Science ...

204 CHAPTER 12 GROUP ACTIONSThe elements of D 4 act on X as functions. The permutation (13)(24) actson vertex 1 by sending it to vertex 3, on vertex 2 by sending it to vertex 4,and so on. It is easy to see that the axioms of a group action are satisfied.In general, if X is any set and G is a subgroup of S X , the group of allpermutations acting on X, then X is a G-set under the group actionfor σ ∈ G and x ∈ X.(σ, x) ↦→ σ(x)Example 3. If we let X = G, then every group G acts on itself by theleft regular representation; that is, (g, x) ↦→ λ g (x) = gx, where λ g is leftmultiplication:e · x = λ e x = ex = x(gh) · x = λ gh x = λ g λ h x = λ g (hx) = g · (h · x).If H is a subgroup of G, then G is an H-set under left multiplication byelements of H.Example 4. Let G be a group and suppose that X = G. If H is a subgroupof G, then G is an H-set under conjugation; that is, we can define an actionof H on G,H × G → G,via(h, g) ↦→ hgh −1for h ∈ H and g ∈ G. Clearly, the first axiom for a group action holds.Observing that(h 1 h 2 , g) = h 1 h 2 g(h 1 h 2 ) −1= h 1 (h 2 gh −12 )h−1 1= (h 1 , (h 2 , g)),we see that the second condition is also satisfied.Example 5. Let H be a subgroup of G and L H the set of left cosets of H.The set L H is a G-set under the action(g, xH) ↦→ gxH.Again, it is easy to see that the first axiom is true. Since (gg ′ )xH = g(g ′ xH),the second axiom is also true.