Contemporary Abstract Algebra9th EditionJoseph Gallian Publisher: Cengage LearningISBN: 9781305657960
Contemporary Abstract Algebra
Solutions for Contemporary Abstract Algebra
Problem 1E:
For each group in the following list, find the order of the group and the order of each element in...Problem 2E:
Let Q be the group of rational numbers under addition and let Q*be the group of nonzero rational...Problem 5E:
Without actually computing the orders, explain why the two elementsin each of the following pairs of...Problem 6E:
In the group Z12 , find a,b,anda+b for each case. a. a=6,b=2 b. a=3,b=8 c. a=5,b=4 Do you see any...Problem 7E:
If a, b, and c are group elements and a=6,b=7 , express (a4c2b4)1 without using negative exponents.Problem 8E:
What can you say about a subgroup of D3 that contains R240 and a reflection F? What can you say...Problem 9E:
What can you say about a subgroup of D4 that contains R270 and a reflection? What can you say about...Problem 10E:
How many subgroups of order 4 does D4 have?Problem 11E:
Determine all elements of finite order in R*, the group of nonzeroreal numbers under multiplication.Problem 12E:
Complete the statement “A group element x is its own inverse if and only if x= ________.”Problem 13E:
For any group elements a and x, prove that xax1=a . This exercise is referred to in Chapter 24.Problem 14E:
Prove that if a is the only element of order 2 in a group, then a lies in the center of the group.Problem 15E:
(1969 Putnam Competition) Prove that no group is the union of two proper subgroups. Does the...Problem 16E:
Let G be the group of symmetries of a circle and R be a rotation of the circle of 2 degrees. What is...Problem 17E:
For each divisor k1 of n, let Uk(n)=xU(n)xmodk=1 .[For example, U3(21)={1,4,10,13,16,19} and...Problem 18E:
Suppose that a is a group element and a6=e . What are the possibilities for a ? Provide reasons for...Problem 20E:
For any group elements a and b, prove that ab=ba .Problem 24E:
Suppose n is an even positive integer and H is a subgroup of Zn .Prove that either every member of H...Problem 25E:
Let n be a positive even integer and let H be a subgroup of Zn of oddorder. Prove that every member...Problem 26E:
Prove that for every subgroup of Dn , either every member of the subgroup is a rotation or exactly...Problem 28E:
Prove that a group with two elements of order 2 that commute musthave a subgroup of order 4.Problem 32E:
Suppose that H is a subgroup of Z under addition and that H contains 250and350 . What are the...Problem 34E:
If H and K are subgroups of G, show that HK is a subgroup of G.(Can you see that the same proof...Problem 35E:
Let G be a group. Show that Z(G)=aGC(a) . [This means the intersection of all subgroups of the form...Problem 37E:
For any group element a and any integer k, show that C(a)C(ak) .Use this fact to complete the...Problem 41E:
Let Sbe a subset of a group and let H be the intersection of all subgroups of G that contain S. a....Problem 42E:
In the group Z, find a. 8,14 ; b. 8,13 ; c. 6,15 ; d. m,n ; e. 12,18,45 . In each part, find an...Problem 43E:
Prove Theorem 3.6. Theorem 3.6 C(a) Is a Subgroup For each a in a group G, the centralizer of a is a...Problem 44E:
If H is a subgroup of G, then by the centralizer C(H) of H we meanthe set xGxh=hx for all hH . Prove...Problem 45E:
Must the centralizer of an element of a group be Abelian? Must the center of a group be Abelian?Problem 46E:
Suppose a belongs to a group and a=5 . Prove that C(a)=C(a3) .Find an element a from some group such...Problem 48E:
In each case, find elements a and b from a group such that a|=|b=2 . a. ab=3 b. ab=4 c. ab=5 Can you...Problem 50E:
Consider the elements A=[0110]andB=[0111] from SL(2,R) . Find A|,|B|,and|AB . Does your answer...Problem 52E:
Give an example of elements a and b from a group such that a has finite order, b has infinite order...Problem 53E:
Consider the element A=[1101] in SL(2,R) . What is the order of A? If we view A=[1101] as a member...Problem 60E:
Compute the orders of the following groups. a. U(3),U(4),U(12) b. U(5),U(7),U(35) c. U(4),U(5),U(20)...Problem 61E:
Let R* be the group of nonzero real numbers under multiplication and let H=xRx2 is rational}. Prove...Problem 62E:
Compute U(4),U(10),andU(40) . Do these groups provide acounter example to your answer to Exercise...Problem 63E:
Find a noncyclic subgroup of order 4 in U(40).Problem 65E:
Let G={[abcd]|a,b,c,dZ} under addition. Let H={[abcd]|G,a+b+c+d=0} . Prove that H is a subgroup of...Problem 67E:
Let H be a subgroup of R under addition. Let K=2aaH .Prove that K is a subgroup of R* under...Problem 68E:
Let G be a group of functions from R to R*, where the operation of G is multiplication of functions....Problem 69E:
Let G=GL(2,R) and H={[a00b]|aandbarenonzerointegers} under the operation of matrix multiplication....Problem 71E:
Let H=a+bia,bR,a2+b2=1 . Prove or disprove that H is a subgroup of C* under multiplication. Describe...Problem 72E:
Let G be a finite Abelian group and let a and b belong to G. Prove that the set a,b=aibji,jZ is a...Problem 74E:
If H and K are nontrivial subgroups of the rational numbers underaddition, prove that HK is...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
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
9th Edition
ISBN: 9781337249560
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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