Contemporary Abstract Algebra9th EditionJoseph Gallian Publisher: Cengage Learning USISBN: 9781337249560
Contemporary Abstract Algebra
Solutions for Contemporary Abstract Algebra
Problem 1E:
Find all generators of Z6,Z8,andZ20 .Problem 2E:
Suppose that a,b,andc are cyclic groups of orders 6, 8, and20, respectively. Find all generators of...Problem 3E:
List the elements of the subgroups 20and10inZ30 . Let a be agroup element of order 30. List the...Problem 4E:
List the elements of the subgroups 3and15inZ18 . Let a be agroup element of order 18. List the...Problem 6E:
What do Exercises 3, 4, and 5 have in common? Try to make a generalization that includes these three...Problem 8E:
Let a be an element of a group and let a=15 . Compute the ordersof the following elements of G. a....Problem 10E:
In Z24 , list all generators for the subgroup of order 8. Let G=aandleta=24 . List all generators...Problem 11E:
Let G be a group and let aG . Prove that a1=a .Problem 12E:
In Z, find all generators of the subgroup 3 . If a has infinite order,find all generators of the...Problem 13E:
In Z24 , find a generator for 2110 . Suppose that a=24 . Finda generator for a21a10 . In general,...Problem 14E:
Suppose that a cyclic group G has exactly three subgroups: G itself,{e}, and a subgroup of order 7....Problem 15E:
Let G be an Abelian group and let H=gG||g divides 12}.Prove that H is a subgroup of G. Is there...Problem 18E:
Let a be a group element and a= . Complete the followingstatement: ai=aj if and only if . . . .Problem 19E:
If a cyclic group has an element of infinite order, how many elementsof finite order does it have?Problem 20E:
Suppose that G is an Abelian group of order 35 and every elementof G satisfies the equation x35=e ....Problem 21E:
Let G be a group and let a be an element of G. a. If a12=e , what can we say about the order of a?...Problem 22E:
Prove that a group of order 3 must be cyclic.Problem 23E:
Let Z denote the group of integers under addition. Is every subgroupof Z cyclic? Why? Describe all...Problem 24E:
For any element a in any group G, prove that a is a subgroup ofC(a) (the centralizer of a).Problem 25E:
If d is a positive integer, d2 , and d divides n, show that the numberof elements of order d in Dn...Problem 26E:
Find all generators of Z. Let a be a group element that has infiniteorder. Find all generators of a...Problem 27E:
Prove that C*, the group of nonzero complex numbers under multiplication,has a cyclic subgroup of...Problem 29E:
List all the elements of order 8 in Z8000000 . How do you know your listis complete? Let a be a...Problem 30E:
Suppose that G is a group with more than one element. If the onlysubgroups of G are {e} and G, prove...Problem 31E:
Let G be a finite group. Show that there exists a fixed positive integern such that an=e for all ain...Problem 32E:
Determine the subgroup lattice for Z12 . Generalize to Zp2q , where pand q are distinct primes.Problem 33E:
Determine the subgroup lattice for Z8 . Generalize to Zpn , where p isa prime and n is some positive...Problem 34E:
Prove that a finite group is the union of proper subgroups if andonly if the group is not cyclic.Problem 35E:
Show that the group of positive rational numbers under multiplicationis not cyclic. Why does this...Problem 36E:
Consider the set {4, 8, 12, 16}. Show that this set is a group undermultiplication modulo 20 by...Problem 37E:
Give an example of a group that has exactly 6 subgroups (includingthe trivial subgroup and the group...Problem 39E:
Suppose that a andb are group elements that commute. If |a| is Finiteand |b| infinite, prove that...Problem 43E:
Suppose that H is a subgroup of a group G and H=10 . If abelongs to G and a6 belongs to H, what are...Problem 45E:
If G is an infinite group, what can you say about the number ofelements of order 8 in the group?...Problem 47E:
For each positive integer n, prove that C*, the group of nonzerocomplex numbers under...Problem 48E:
Prove or disprove that H=nZn is divisible by both 8 and 10}is a subgroup of Z. What happens if...Problem 60E:
Given the fact that U(49) is cyclic and has 42 elements, deduce thenumber of generators that U(49)...Problem 64E:
Prove that U(2n)(n3) is not cyclic.Problem 66E:
Prove that Zn has an even number of generators if n2 . What doesthis tell you about (n) ?Problem 67E:
If a5=12 , what are the possibilities for |a|? If a4=12 , what arethe possibilities for |a|?Problem 68E:
Suppose that x=n . Find a necessary and sufficient condition on rand s such that xrkxs .Problem 69E:
Let a be a group element such that a=48 . For each part, find adivisor k of 48 such that a21=ak;...Problem 71E:
Suppose that |a| and |b| are elements of a group and a andb commute.If a=5andb=16 , prove that ab=80...Problem 74E:
If n is an even integer prove that (2n)=2(n) .Browse All Chapters of This Textbook
Chapter 0 - PreliminariesChapter 1 - Introduction To GroupsChapter 2 - GroupsChapter 3 - Finite Groups; SubgroupsChapter 4 - Cyclic GroupsChapter 5 - Permutation GroupsChapter 6 - IsomorphismsChapter 7 - Cosets And Lagrange’s TheoremChapter 9 - Normal Subgroups And Factor GroupsChapter 10 - Group Homomorphisms
Chapter 12 - Introduction To RingsChapter 13 - Integral DomainsChapter 14 - Ideals And Factor RingsChapter 18 - Divisibility In Integral DomainsChapter 20 - Extension FieldsChapter 26 - Generators And RelationsChapter 28 - Frieze Groups And Crystallographic GroupsChapter 30 - Cayley Digraphs Of Groups
Sample Solutions for this Textbook
We offer sample solutions for Contemporary Abstract Algebra homework problems. See examples below:
Chapter 0, Problem 1EGiven: An equilateral triangle. Calculation: Three rotations at 0, 120 and 240 gives the symmetric...Chapter 2, Problem 1EChapter 3, Problem 1EChapter 4, Problem 1EChapter 5, Problem 1EGiven information: Concept used: Isomorphism: - A homomorphism ϕ from G into G¯ is said to be an...Chapter 7, Problem 1EChapter 9, Problem 1E
More Editions of This Book
Corresponding editions of this textbook are also available below:
Contemporary Abstract Algebra.
6th Edition
ISBN: 9780618514717
EBK CONTEMPORARY ABSTRACT ALGEBRA
8th Edition
ISBN: 9780100453074
Contemporary Abstract Algebra
8th Edition
ISBN: 9781133599708
Contemporary Abstract Algebra
8th Edition
ISBN: 9781285402734
Contemporary Abstract Algebra
8th Edition
ISBN: 9788131520741
Student Solutions Manual For Gallian's Contemporary Abstract Algebra, 8th
8th Edition
ISBN: 9781133608530
EBK CONTEMPORARY ABSTRACT ALGEBRA
8th Edition
ISBN: 8220100453076
Contemporary Abstract Algebra, 7th Edition
7th Edition
ISBN: 9780547165097
Student Solutions Manual For Gallian's Contemporary Abstract Algebra, 7th (students Solutions Manual)
7th Edition
ISBN: 9780547165394
Student Solutions Manual for Gallian's Contemporary Abstract Algebra, 9th
9th Edition
ISBN: 9781305657977
Contemporary Abstract Algebra
9th Edition
ISBN: 9781305657960
Bundle: Contemporary Abstract Algebra, 9th + Student Solutions Manual
9th Edition
ISBN: 9781337501590
EBK CONTEMPORARY ABSTRACT ALGEBRA
9th Edition
ISBN: 8220101434852
EBK CONTEMPORARY ABSTRACT ALGEBRA
9th Edition
ISBN: 9781305887855
Contemporary Abstract Algebra
5th Edition
ISBN: 9780618122141
Contemporary Abstract Algebra
10th Edition
ISBN: 9781000337358
CONTEMPORARY ABSTRACT ALGEBRA
10th Edition
ISBN: 9780367651787
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