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Unformatted text preview: Exam 3 Review Sheet Solutions Math 4023 Section 1 The third exam will be on Monday, November 21, 2011. It will cover Sections 5.1 - 5.5. 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: ² S n denotes the set of all permutations of the set f 1 ;:::;n g of integers from 1 to n . The cardinality of S n is j S n j = n !. ² Know how to represent permutations in the two rowed notation, and how to multiply permu- tations using this notation. ² Know what a cycle of length r is. A cycle of length 2 is a transposition . ² Know what it means to say that two permutations … and are disjoint . ² Disjoint permutations commute. (Theorem 5.1.2, Page 209) ² Know how to compute the cycle decomposition of permutations in S n . ² 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. ² Know what is meant by the order of a permutation: o ( … ) is the smallest positive integer k such that … k = id. That is, o ( … ) is the order of … as an element of the group S n ² The order of an r-cycle is r . ² Know how to compute the order of a permutation from the cycle structure: If … = ¿ 1 ¿ 2 ¢¢¢ ¿ k is a product of disjoint cycles, then the order of … is the least common multiple of the lengths of the cycles ¿ 1 , ::: , ¿ k . ² 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 is always even or always odd. is even if it is a product of an even number of transpositions; is odd if it is a product of an odd number of transpositions. ² 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 ) : 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. ² The alternating group A n S n is the subgroup of all even permutation. The order of A n is j A n j = n ! = 2. ² Know what we mean by the symmetry group Sym ( X ) of a set X in the plane (or three space). ² Know how to represent the symmetries of a polygon P by means of permutations of the vertices. (Page 215) 1 Exam 3 Review Sheet Solutions Math 4023 Section 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 2 k…=n radians about the center of the polygon, and the last n entries are re ections....
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This document was uploaded on 12/28/2011.
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