The document discusses group actions, defining a left action of a group g on a set x, and introduces concepts such as g-equivalence, orbits, fixed point sets, and stabilizer subgroups. It includes a proposition regarding g-equivalence as an equivalence relation and presents Burnside's Counting Theorem to illustrate how many distinct ways vertices of a square can be colored using three colors, accounting for equivalent colorings. An example using the dihedral group D4 and the set of colorings is provided to demonstrate the application of the theorem.

1. Group Actions Amanda Trommater

2. Definition Let X be a set and G be a group. A (left) action of G on X is a map by , where ex = x for all x X ; ( g 1 g 2 ) x = g 1 ( g 2 x ) for all x X and all g 1 ,g 2 G .

3. Example G =D 4 and the set of vertices of a square X ={1,2,3,4}

4. G-equivalence If G acts on a set X and x, y X , then x is said to be G-equivalent to y if there exists a g G such that gx=y Proposition 12.2: Let X be a G -set. Then G-equivalence is an equivalence relation on X .

5. Orbits If X is a G-set, each partition of X associated with G-equivalence is an orbit

6. Fixed Point Set Let g be an element of G . The fixed point set of g in X is the set of all x X such that gx=x .

7. Stabilizer Subgroup The group of elements g that fix a given x X . G x is a subgroup of G by Proposition 12.2 Theorem 12.3

8. Burnside’s Counting Theorem How many ways can you color the vertices of a square with 3 colors? May suspect 3 4 , but some colorings are equivalent

9. Burnside’s Theorem Let G be a finite group acting on a set X and let k denote the number of orbits of X . Then

10. How many ways can the vertices of a square be colored using 3 colors? G = D 4 ={(1), (13), (24), (12)(34), (14)(23), (13)(24),(1234), (1423)} X ={1,2,3,4} Let Y ={B,W,R}, the set of different colorings (black, white, and red) Map