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Unformatted text preview: Exam 3 Review Sheet Math 4023 Section 1 The third exam will be on Monday, November 22, 2010. It will cover Sections III.9, V.1, V.2, V.3, V.4, V.5 plus the handout on Burnside coloring arguments. Of course, the material is cumulative, and the listed sections depend on earlier sections, which it is assumed that you still know. Following are some of the concepts and results you should know: Know what it means to say that a group G acts on a set X (Deflnition 9.2, Page 118). Namely, each element g of G determines a permutation of X , denoted by x 7! gx , in such a way that: (a) ex = x for all x 2 X , where e is the identity of G . That is the permutation of X determined by e is the identity permutation. (b) ( gh ) x = g ( hx ) for all g , h 2 G and x 2 X . That is, the multiplication of g and h in G corresponds to the permutation of X that is the composition of the permutation determined by h and that determined by g . If G is a group acting on a set X and a 2 X , know what the orbit of a under the action of G is. Namely, Orb( a ) = f ga : g 2 G g : That is, the orbit of a consists of all of the elements of X that are the image of a under one of the permutations of X determined by an element of G . Know what the stabilizer of a , denoted G a , is: G a = f g 2 G : ga = a g : That is the stabilizer of a consists of all the group elements g that do not move a under the permutation of X determined by g . If G acts on a set X , then the orbits of this action form a partition of X . If G acts on a set X , then the stabilizer G a of each element a 2 X is a subgroup of G . Know the OrbitStabilizer theorem for group actions (Proposition 9.11, Page 121): If G acts on X and a 2 X is any element of X , then j Orb( a ) j = [ G : G a ] = j G j j G a j = the number of cosets of G a in G: Another way to state the same thing is: j Orb( a ) jj G a j = j G j : If G is a group acting on the set X and g 2 G , know what the flxed set of g , denoted X g is: X g = f x 2 X : gx = x g . That is, X g is the set of elements of X that are not moved by the permutation of X determined by g . Know the Burnside counting theorem (Theorem 9.15, Page 122: If G is a flnite group acting on a flnite set X , then the number N of orbits in X under this action of G is N = 1 j G j X g 2 G j X g j where X g is the number of elements in X that are flxed by g . A convenient way to think about this theorem is that it says that the number of orbits is the average (as g varies over G ) of the number of elements of X flxed by an element g of G . 1 Exam 3 Review Sheet Math 4023 Section 1 Know how to use Burnsides theorem to compute the number of distinct patterns in the colorings of a given set of points in the plane, using the action of the symmetry group of the points....
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 Fall '09

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