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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.

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

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Transcript

  • 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

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