SOLUTIONS MANUAL
, v
CONTEMPORARY ABSTRACT ALGEBRA 11TH EDITION
INSTRUCTOR SOLUTIONS MANUAL
CONTENTS
0 Introduction to Groups 5
1 Groups 7
2 Finite Groups; Subgroups 11
3 Cyclic Groups 18
4 Permutation Groups 25
5 Isomorphisms 31
6 Cosets and Lagrange’s Theorem 37
7 External Direct Products 44
8 Normal Subgroups and Factor Groups 50
9 Group Homomorphisms 56
10 Fundamental Theorem of Finite Abelian Groups 62
11 Introduction to Rings 66
12 Integral Domains 71
13 Ideals and Factor Rings 77
14 Ring Homomorphisms 84
15 Polynomial Rings 91
16 Factorization of Polynomials 97
17 Divisibility in Integral Domains 102
,vi
Fields
19 Extension Fields 106
20 Algebraic Extensions 111
21 Finite Fields 116
22 Geometric Constructions 121
Special Topics
23 Sylow Theorems 123
24 Finite Simple Groups 129
25 Generators and Relations 133
26 Symmetry Groups 136
27 Symmetry and Counting 138
28 Cayley Digraphs of Groups 140
29 Introduction to Algebraic Coding Theory 143
30 An Introduction to Galois Theory 147
31 Cyclotomic Extensions 150
, 5
CHAPTER 1
Introduction to Groups
1. Three rotations: 0◦, 120◦, 240◦, and three reflections across lines from vertices to
midpoints of opposite sides.
2. Let R = R120, R2 = R240, F be a reflection across a vertical axis, F ′ = RF , and
F ′′ = R2F
R0 R R F′ F ′′
2
F
R0 R0 R R 2
F F′ F ′′
R R R0 F F ′′
2 ′
R F
R2 R2 R0 R F ′′ F F′
F F F ′ R0 R2
′′
F R
F ′ F ′ F F ′′ R R0 R2
F ′′ F ′′ F ′ F R2 R R0
3. a. V b. R 270 c. R0 d. R0, R180, H, V, D, D′ e. none
4. Five rotations: 0◦, 72◦, 144◦, 216◦, 288◦, and five reflections across lines from
vertices to midpoints of opposite sides.
5. Dn has n rotations of the form k(360◦/n), where k = 0, . . . , n — 1. In addition, D n
has n reflections. When n is odd, the axes of reflection are the lines from the vertices
to the midpoints of the opposite sides. When n is even, half of the axes of reflection
are obtained by joining opposite vertices; the other half, by joining midpoints of
opposite sides.
6. A nonidentity rotation leaves only one point fixed – the center of rotation. A
reflection leaves the axis of reflection fixed. A reflection followed by a different
reflection would leave only one point fixed (the intersection of the two axes of
reflection), so it must be a rotation.
7. A rotation followed by a rotation either fixes every point (and so is the identity) or
fixes only the center of rotation. However, a reflection fixes a line.
8. In either case, the set of points fixed is some axis of reflection.
9. Observe that 1 · 1 = 1; 1(—1) = —1; (—1)1 = —1; (—1)(—1) = 1. These relationships
also hold when 1 is replaced by a “rotation” and —1 is replaced by a “reflection.”
10. Reflection.
11. Thinking geometrically and observing that even powers of elements of a dihedral
group do not change orientation, we note that each of a, b and c appears an even
number of times in the expression. So, there is no change in orientation. Thus, the
expression is a rotation. Alternatively, as in Exercise 9, we associate each of a, b and
c with 1 if they are rotations and —1 if they are reflections and we observe that in
the product a2b4ac5a3c the terms involving a represent six 1s or six —1s, the term b4
represents four 1s or four —1s, and the terms involving c represent six 1s or six —1s.
Thus the product of all the 1s and —1s is 1. So the expression is a rotation.
, v
CONTEMPORARY ABSTRACT ALGEBRA 11TH EDITION
INSTRUCTOR SOLUTIONS MANUAL
CONTENTS
0 Introduction to Groups 5
1 Groups 7
2 Finite Groups; Subgroups 11
3 Cyclic Groups 18
4 Permutation Groups 25
5 Isomorphisms 31
6 Cosets and Lagrange’s Theorem 37
7 External Direct Products 44
8 Normal Subgroups and Factor Groups 50
9 Group Homomorphisms 56
10 Fundamental Theorem of Finite Abelian Groups 62
11 Introduction to Rings 66
12 Integral Domains 71
13 Ideals and Factor Rings 77
14 Ring Homomorphisms 84
15 Polynomial Rings 91
16 Factorization of Polynomials 97
17 Divisibility in Integral Domains 102
,vi
Fields
19 Extension Fields 106
20 Algebraic Extensions 111
21 Finite Fields 116
22 Geometric Constructions 121
Special Topics
23 Sylow Theorems 123
24 Finite Simple Groups 129
25 Generators and Relations 133
26 Symmetry Groups 136
27 Symmetry and Counting 138
28 Cayley Digraphs of Groups 140
29 Introduction to Algebraic Coding Theory 143
30 An Introduction to Galois Theory 147
31 Cyclotomic Extensions 150
, 5
CHAPTER 1
Introduction to Groups
1. Three rotations: 0◦, 120◦, 240◦, and three reflections across lines from vertices to
midpoints of opposite sides.
2. Let R = R120, R2 = R240, F be a reflection across a vertical axis, F ′ = RF , and
F ′′ = R2F
R0 R R F′ F ′′
2
F
R0 R0 R R 2
F F′ F ′′
R R R0 F F ′′
2 ′
R F
R2 R2 R0 R F ′′ F F′
F F F ′ R0 R2
′′
F R
F ′ F ′ F F ′′ R R0 R2
F ′′ F ′′ F ′ F R2 R R0
3. a. V b. R 270 c. R0 d. R0, R180, H, V, D, D′ e. none
4. Five rotations: 0◦, 72◦, 144◦, 216◦, 288◦, and five reflections across lines from
vertices to midpoints of opposite sides.
5. Dn has n rotations of the form k(360◦/n), where k = 0, . . . , n — 1. In addition, D n
has n reflections. When n is odd, the axes of reflection are the lines from the vertices
to the midpoints of the opposite sides. When n is even, half of the axes of reflection
are obtained by joining opposite vertices; the other half, by joining midpoints of
opposite sides.
6. A nonidentity rotation leaves only one point fixed – the center of rotation. A
reflection leaves the axis of reflection fixed. A reflection followed by a different
reflection would leave only one point fixed (the intersection of the two axes of
reflection), so it must be a rotation.
7. A rotation followed by a rotation either fixes every point (and so is the identity) or
fixes only the center of rotation. However, a reflection fixes a line.
8. In either case, the set of points fixed is some axis of reflection.
9. Observe that 1 · 1 = 1; 1(—1) = —1; (—1)1 = —1; (—1)(—1) = 1. These relationships
also hold when 1 is replaced by a “rotation” and —1 is replaced by a “reflection.”
10. Reflection.
11. Thinking geometrically and observing that even powers of elements of a dihedral
group do not change orientation, we note that each of a, b and c appears an even
number of times in the expression. So, there is no change in orientation. Thus, the
expression is a rotation. Alternatively, as in Exercise 9, we associate each of a, b and
c with 1 if they are rotations and —1 if they are reflections and we observe that in
the product a2b4ac5a3c the terms involving a represent six 1s or six —1s, the term b4
represents four 1s or four —1s, and the terms involving c represent six 1s or six —1s.
Thus the product of all the 1s and —1s is 1. So the expression is a rotation.
