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)?
6. 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.
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_forwardMarco sorted these solid figures into two groups. What is true about the groups? Group 1 Group 2 1) All figures in Group 2 appear to have at least one triangular face. All figures in Group 2 are triangular prisms. All figures in Group 1 are rectangular prisms. All figures in Group 1 appear to have at least one square face. Done ->arrow_forward
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