Contemporary Abstract Algebra
9th Edition
ISBN: 9781305657960
Author: Joseph Gallian
Publisher: Cengage Learning
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Textbook Question
Chapter 5, Problem 50E
Viewing the members of
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The permutations p = (3 5 4), q = (1 5 2 3) and r = (1 5 2) are
elements of S5.
(a) Find each of the following as a permutation in cycle form:
wold wode ei doang
poq,
poqop-¹, q².
(b) State the order and parity of each of po q and q².
(c) Determine, in cycle form, two permutations in S5 that conjugate p
to r.
Viewing the members of D5 as a group of permutations of a regularpentagon with consecutive vertices labeled 1, 2, 3, 4, 5, what geometricsymmetry corresponds to the permutation (14253)? Whichsymmetry corresponds to the permutation (25)(34)?
The number 52 is written as a sum of three natural numbers
52 = a +b+c
(the triple (a, b, c) is ordered; e.g., the decompositions 52 = 14 + 15 + 23 and 52 = 15 + 23 + 14 are different.
Also, assume that all the decompositions have equal probability.)
Given that there exists a triangle with sides a, b, and c, what is the probability that this triangle is isosceles?
Chapter 5 Solutions
Contemporary Abstract Algebra
Ch. 5 - Let [123456213546]and=[123456612435] . Compute...Ch. 5 - Let [1234567823451786]and=[1234567813876524] ....Ch. 5 - Write each of the following permutations as a...Ch. 5 - Find the order of each of the following...Ch. 5 - What is the order of each of the following...Ch. 5 - What is the order of each of the following...Ch. 5 - What is the order of the product of a pair of...Ch. 5 - Determine whether the following permutations are...Ch. 5 - What are the possible orders for the elements of...Ch. 5 - Show that A8 contains an element of order 15.
Ch. 5 - Find an element in A12 of order 30.Ch. 5 - Show that a function from a finite set S to itself...Ch. 5 - Prob. 13ECh. 5 - Suppose that is a 6-cycle and is a 5-cycle....Ch. 5 - Prob. 15ECh. 5 - If is even, prove that 1 is even. If is odd,...Ch. 5 - Prob. 17ECh. 5 - In Sn , let be an r-cycle, an s-cycle, and a...Ch. 5 - Let and belong to Sn . Prove that is even if...Ch. 5 - Associate an even permutation with the number +1...Ch. 5 - Complete the following statement: A product of...Ch. 5 - What cycle is (a1a2an)1 ?Ch. 5 - Show that if H is a subgroup of Sn , then either...Ch. 5 - Suppose that H is a subgroup of Sn of odd order....Ch. 5 - Give two reasons why the set of odd permutations...Ch. 5 - Let and belong to Sn . Prove that 11 is an...Ch. 5 - Prob. 27ECh. 5 - How many elements of order 5 are in S7 ?Ch. 5 - Prob. 29ECh. 5 - Prove that (1234) is not the product of 3-cycles....Ch. 5 - Let S7 and suppose 4=(2143567) . Find . What are...Ch. 5 - My mind rebels at stagnation. Give me problems,...Ch. 5 - Let (a1a2a3a4)and(a5a6) be disjoint cycles in S10...Ch. 5 - If and are distinct 2-cycles, what are the...Ch. 5 - Prob. 35ECh. 5 - Let =(1,3,5,7,9,8,6)(2,4,10) . What is the...Ch. 5 - Prob. 37ECh. 5 - Let H=S5(1)=1and(3)=3 . Prove that H is a...Ch. 5 - In S4 , find a cyclic subgroup of order 4 and a...Ch. 5 - In S3 , find elements and such that...Ch. 5 - Find group elements and in S5 such that...Ch. 5 - Represent the symmetry group of an equilateral...Ch. 5 - Prove that Sn is non-Abelian for all n3 .Ch. 5 - Prove that An is non-Abelian for all n4 .Ch. 5 - For n3 , let H=bSn(1)=1 or 2 and (2)=1or2 .Prove...Ch. 5 - Show that in S7 , the equation x2=(1234) has no...Ch. 5 - If (ab) and (cd) are distinct 2-cycles in Sn ,...Ch. 5 - Let and belong to Sn . Prove that 1 and are...Ch. 5 - Viewing the members of D4 as a group of...Ch. 5 - Viewing the members of D5 as a group of...Ch. 5 - Prob. 51ECh. 5 - Prob. 52ECh. 5 - Show that A5 has 24 elements of order 5, 20...Ch. 5 - Find a cyclic subgroup of A8 that has order 4....Ch. 5 - Prob. 55ECh. 5 - Prob. 56ECh. 5 - Show that every element in An for n3 can be...Ch. 5 - Show that for n3,Z(Sn)=[] .Ch. 5 - Prob. 59ECh. 5 - Use the Verhoeff check-digit scheme based on D5 to...Ch. 5 - Prob. 61ECh. 5 - (Indiana College Mathematics Competition) A...Ch. 5 - Prob. 63ECh. 5 - Find five subgroups of S5 of order 24.Ch. 5 - Why does the fact that the orders of the elements...Ch. 5 - Let a belong to Sn . Prove that divides n!Ch. 5 - Encrypt the message ATTACK POSTPONED using the...Ch. 5 - The message VAADENWCNHREDEYA was encrypted using...
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- Exercises 1. Express each permutation as a product of disjoint cycles and find the orbits of each permutation. a. b. c. d. e. f. g. h.arrow_forwardAn organization wants to assemble a cluster of n computers using wires and hubs. Each computer must have exactly one wire attached to it, while each hub can have up to three wires attached. There must be a path of wires between every pair of computers in the cluster. Based on this information, [(n – 2)/3] hubs are sufficient for the organization to assemble the cluster of n computers. True Falsearrow_forward6. The group of symmetries of the tetrahedron has the following permutations: e = (1)(2)3(4); r, = (1)(234); P, = (1)(243); r, = (2{143); P; = (2)(134); r, = (3)(124); p, = (3)(142); = (4)(132); p. = (4(123); o, = (14)(23); o, = (24)(13); a, = (34)(12); Show that H = {e, r,, Pa) is a subgroup of this group by showing closure under composition. Find all the right cosets gH, of H.arrow_forward
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