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# Group Actions

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Chapter presentation project by Amanda Trommater for MAT 361 (Modern Algebra) at Franklin College, Fall 2009.

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### Group Actions

1. 1. Group Actions Amanda Trommater
2. 2. Definition <ul><li>Let X be a set and G be a group. A (left) action of G on X is a map by , where </li></ul><ul><ul><li>ex = x for all x X ; </li></ul></ul><ul><ul><li>( g 1 g 2 ) x = g 1 ( g 2 x ) for all x X and all g 1 ,g 2 G . </li></ul></ul>
3. 3. Example <ul><li>G =D 4 and the set of vertices of a square X ={1,2,3,4} </li></ul>
4. 4. G-equivalence <ul><li>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 </li></ul><ul><li>gx=y </li></ul><ul><li>Proposition 12.2: Let X be a G -set. Then G-equivalence is an equivalence relation on X . </li></ul>
5. 5. Orbits <ul><li>If X is a G-set, each partition of X associated with G-equivalence is an orbit </li></ul>
6. 6. Fixed Point Set <ul><li>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 . </li></ul>
7. 7. Stabilizer Subgroup <ul><li>The group of elements g that fix a given x X . </li></ul><ul><li>G x is a subgroup of G by Proposition 12.2 </li></ul><ul><li>Theorem 12.3 </li></ul>
8. 8. Burnside’s Counting Theorem <ul><li>How many ways can you color the vertices of a square with 3 colors? </li></ul><ul><ul><li>May suspect 3 4 , but some colorings are equivalent </li></ul></ul>
9. 9. Burnside’s Theorem <ul><li>Let G be a finite group acting on a set X and let k denote the number of orbits of X . Then </li></ul>
10. 10. How many ways can the vertices of a square be colored using 3 colors? <ul><li>G = D 4 ={(1), (13), (24), (12)(34), (14)(23), (13)(24),(1234), (1423)} </li></ul><ul><li>X ={1,2,3,4} </li></ul><ul><li>Let Y ={B,W,R}, the set of different colorings (black, white, and red) </li></ul><ul><li>Map </li></ul>