Know how to represent permutations in the two rowed notation, and how to multiply permutations using this notation.

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1 The third exam will be on Monday, November 21, It will cover Sections 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: S n denotes the set of all permutations of the set {1,..., n} of integers from 1 to n. cardinality of S n is S n = n!. The Know how to represent permutations in the two rowed notation, and how to multiply permutations using this notation. Know what a cycle of length r is. A cycle of length 2 is a transposition. Know what it means to say that two permutations π and σ are disjoint. Disjoint permutations commute. (Theorem 5.1.2, Page 209) Know how to compute the cycle decomposition of permutations in S n. Know how to go back and forth between two rowed notation for permutations and cycle decompositions. Know how to multiply permutations given in either format and express the result in either two rowed or cycle notation. Know what is meant by the order of a permutation: o(π) is the smallest positive integer k such that π k = id. That is, o(π) is the order of π as an element of the group S n The order of an r-cycle is r. Know how to compute the order of a permutation from the cycle structure: If π = τ 1 τ 2 τ k is a product of disjoint cycles, then the order of π is the least common multiple of the lengths of the cycles τ 1,..., τ k. A transposition is a cycle of length 2. Every permutation is a product of transpositions. The number of transpositions in such a product for a permutation σ is always even or always odd. σ is even if it is a product of an even number of transpositions; σ is odd if it is a product of an odd number of transpositions. An r-cycle (j 1, j 2,..., j r ) is an even permutation if r is odd and it is an odd permutation if r is even. This follows from the factorization (j 1 j 2... j r ) = (j 1 j r )(j 1 j r 1 ) (j 1 j 2 ). The product of two even permutations is even, the product of two odd permutations is even, and the product of an even and an odd permutation is odd. The alternating group A n S n is the subgroup of all even permutation. The order of A n is A n = n!/2. Know what we mean by the symmetry group Sym(X) of a set X in the plane (or three space). Know how to represent the symmetries of a polygon P by means of permutations of the vertices. (Page 215) 1

2 The symmetry group of a regular polygon with n-sides is the dihedral group of degree n D n = { e, a,..., a n 1, b, ab,..., a n 1 b }, where a k is the (counterclockwise) rotation by 2kπ/n radians about the center of the polygon, and the last n entries are reflections. The subgroup C n = { e, a,..., a n 1} of D n is the rotation group or order n. If the symmetry group of a plane figure X is finite, then it is either D n or C n for some n. (Page 218) Know what it means to say that a group G acts on a set X (Definition 5.4.1, Page 222). Namely, each element g of G determines a permutation of X, denoted by x gx, in such a way that: (a) ex = x for all x 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 G and x 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 X, know what the orbit of a under the action of G is. Namely, Orb(a) = {ga : 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 Stab(a), is: Stab(a) = {g G : ga = a}. 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 Stab(a) of each element a X is a subgroup of G. Know the Orbit-Stabilizer theorem for group actions (Theorem 5.4.3, Page 223): If G acts on X and a X is any element of X, then Orb(a) = (G : Stab(a)) = Another way to state the same thing is: G = the number of cosets of Stab(a) in G. Stab(a) Orb(a) Stab(a) = G. If G is a group acting on the set X and g G, know what the fixed set of g, denoted Fix(g) is: Fix(g) = {x X : gx = x}. That is, Fix(g) is the set of elements of X that are not moved by the permutation of X determined by g. 2

3 Know the Burnside counting theorem (Theorem 5.4.4, Page 224: If G is a finite group acting on a finite set X, then the number k of orbits in X under this action of G is k = 1 G Fix(g) g G where Fix(g) is the number of elements in X that are fixed 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 fixed by an element g of G. Know how to use Burnside s 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. Review Exercises Be sure that you know how to do all assigned homework exercises. The following are a few supplemental exercises similar to those already assigned as homework. 1. Let α = ( ) and β = ( 1 2 ) ( ). (Note that we are using the cycle notation for α and β which we are assuming are in S 6. (a) Write α and β in two rowed notation. (b) Express α 1 and α 2 as products of disjoint cycles. (c) Write β as a product of 2 cycles. Is β even or odd? (d) Let θ = στ where σ and τ are disjoint cycles of length 9 and 6, respectively. Find the smallest positive integer s such that θ s is the identity permutation. 2. Write each of the following permutations as a product of disjoint cycles. (Remember that a single cycle qualifies.) (a) (b) (c) ( 1 2 ) ( 1 3 ) ( 1 4 ) (d) ( 1 3 ) 1 ( 2 4 ) ( ) 1 (e) ( ) ( ) ( 1 3 ) (f) ( ) Assume that σ = and τ = each of the following elements of S 4 : (a) στ (b) τσ (c) σ 2 (d) τ 2 (e) τ 3 (f) τ 4 (g) σ 1 (h) τ are permutations in S Compute 3

4 4. Write each of the following permutations as a single cycle or a product of disjoint cycles. (a) (b) (c) (d) In S 10, let α = ( ), β = ( ), γ = ( ), and let σ = αβγ. Write σ as a product of disjoint cycles, and use this to find its order and its inverse. Is σ even or odd? 6. Count the number of ways that a 3 3 grid of squares can be colored red or black (adjacent squares are allowed to have the same color). We will assume that two colorings are the same when one can be obtained from the other by clockwise rotation by 90, 180, or 270. It may be helpful to use the following two grids for visualization purposes The grid on the right is obtained from the one on the left by rotation ρ by 90, so that ρ can be identified with the following permutation of the nine small squares: ρ = ( ) ( ) ( 5 ) The following represents a particular coloring of the grid and the coloring obtained by 90 clockwise rotation. Thus these two colorings will be considered the same. R B B R R B B R R B R R R R B R B B 7. Find the number of different regular pentagons with vertices colored red, white, or blue. 8. A wheel is divided evenly into four compartments. Each compartment can be painted red, white, or blue. The back of the wheel is black. How many different such color wheels are there? 9. A rectangular necktie is divided evenly into five bands and each band may be colored green, red, or blue. How many different neckties are there? 10. A rectangular electrical relay box appears as shown below: 4

5 If each wire may be red or black, how many such boxes, taking symmetry into account, are there? If each wire may be red, black, or white, how many such boxes are there? 11. Think of the set of all six-digit binary words that is, strings of length 6 composed of 0 s and 1 s. For example: In some applications, two six-digit words are considered equivalent if one can be obtained from the other by applying the cyclic permutation σ: a 1 a 2 a 3 a 4 a 5 a 6 a 6 a 1 a 2 a 3 a 4 a 5 a number of times. For example, is equivalent to since the second string is obtained from the first by applying σ 3 times. Find the number of such non-equivalent words. 5