A probabilistic proof of a lemma that is not Burnside's
aa r X i v : . [ m a t h . HO ] J u l A Probabilistic Proof of aLemma that is not Burnside’s
Vincent Vatter
Department of MathematicsUniversity of FloridaGainesville, Florida USA
If the group G acts on the set X , we define the orbit of the element x ∈ X asorb( x ) = { gx : g ∈ G } . The orbits partition X , and the orbit-counting lemma (see Neumann [2] for the history ofits name) shows how to compute the number of orbits. Bogart [1] gave a proof of this resultusing only multisets and the product rule. We give a probabilistic version of his proof. The Orbit-Counting Lemma.
Suppose the finite group G acts on the finite set X andlet fix( g ) = { x ∈ X : gx = x } . Then the number of orbits of X is | G | X g ∈ G | fix( g ) | . Proof.
Choose, in order and each uniformly at random, an element g ∈ G , an orbit of X ,and an element y of this orbit. Because gy = x for some x ∈ X ,1 = X x ∈ X Pr (cid:2) gy = x (cid:3) = X x ∈ X Pr (cid:2) y ∈ orb( x ) (cid:3) · Pr (cid:2) gy = x (cid:12)(cid:12) y ∈ orb( x ) (cid:3) . If y ∈ orb( x ), then there is some k ∈ G such that kx = y , so the mapping h k − h − is a bijection between { h ∈ G : hy = x } and { h ∈ G : hx = x } . By applying this(measure-preserving) mapping, it follows that, for every x ∈ X ,Pr (cid:2) gy = x (cid:12)(cid:12) y ∈ orb( x ) (cid:3) = Pr (cid:2) gx = x (cid:12)(cid:12) y ∈ orb( x ) (cid:3) = Pr (cid:2) gx = x (cid:3) . Also, since the orbit y was chosen from was itself chosen uniformly at random,1 X x ∈ X Pr (cid:2) gx = x (cid:3) = 1 . This implies that the number of orbits is equal to X x ∈ X Pr (cid:2) gx = x (cid:3) = 1 | G | (cid:12)(cid:12)(cid:8) ( g, x ) ∈ G × X : gx = x (cid:9)(cid:12)(cid:12) = 1 | G | X g ∈ G | fix( g ) | , as claimed. References [1] Bogart, K. (1991). An obvious proof of Burnside’s lemma.
Amer. Math. Monthly . (10):927–928.[2] Neumann, P. (1979). A lemma that is not Burnside’s. Math. Sci. (2): 133–141. This note appeared as a “MathBit” in
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