Abstract Algebra Theory and Applications - Computer Science ...

12.3 BURNSIDE’S COUNTING THEOREM 213Proposition 12.8 Let G be a permutation group of X and ˜X the set offunctions from X to Y . Then there exists a permutation group ˜G actingon ˜X, where ˜σ ∈ ˜G is defined by ˜σ(f) = f ◦ σ for σ ∈ G and f ∈ ˜X.Furthermore, if n is the number of cycles in the cycle decomposition of σ,then | ˜X σ | = |Y | n .Proof. Let σ ∈ G and f ∈ ˜X. Clearly, f ◦ σ is also in ˜X. Suppose thatg is another function from X to Y such that ˜σ(f) = ˜σ(g). Then for eachx ∈ X,f(σ(x)) = ˜σ(f)(x) = ˜σ(g)(x) = g(σ(x)).Since σ is a permutation of X, every element x ′ in X is the image of some xin X under σ; hence, f and g agree on all elements of X. Therefore, f = gand ˜σ is injective. The map σ ↦→ ˜σ is onto, since the two sets are the samesize.Suppose that σ is a permutation of X with cycle decomposition σ =σ 1 σ 2 · · · σ n . Any f in ˜X σ must have the same value on each cycle of σ.Since there are n cycles and |Y | possible values for each cycle, | ˜X σ | = |Y | n .□Example 12. Let X = {1, 2, . . . , 7} and suppose that Y = {A, B, C}. If gis the permutation of X given by (13)(245) = (13)(245)(6)(7), then n = 4.Any f ∈ F g must have the same value on each cycle in g. There are |Y | = 3such choices for any value, so |F g | = 3 4 = 81.Example 13. Suppose that we wish to color the vertices of a square usingfour different colors. By Proposition 12.8, we can immediately decide thatthere are18 (44 + 4 1 + 4 2 + 4 1 + 4 2 + 4 2 + 4 3 + 4 3 ) = 55possible ways.x 1x 2.x n✲✲✲f✲ f(x 1 , x 2 , . . . , x n )Figure 12.2. A switching function of n variables