A Hungerford’s Algebra
Solutions Manual
Volume I: Introduction through Chapter IV
James Wilson
D4
⟨a 2, b⟩ ⟨a⟩ ⟨a ,2ab⟩
I
⟨b⟩ ⟨a2 b⟩ ⟨a2 ⟩ ⟨ab⟩ ⟨a 3b⟩
0
0 = C0(G) ≤ C1(G) ≤ ≤ Cn−1(G) ≤Cn(G) = G
II 0 = Gn ≤ Gn−1 ≤ ≤ G1 ≤ G0 = G
0 = Γn+1G≤ Γn G ≤ ≤ Γ2G ≤ Γ 1G = G
−
Commutative Local
Ring Ring
Field
−−−− −−− −
Unique
Integral
Factorization
Domain
III Ring Unital
Domain
Principal
Ring Ideal −−−−−−−−−
−
Domain
Skew
Field
Principal
Ideal Euclidean
Ring Domain
··········
Euclidean
Ring
= ≤
0 A B C 0
··········
IV
= ≤
0 A′ B′ C′ 0
··········
,−−−−− −−− −−−
−−−−−
−
−−−−−−−−−
−−−
−−−−−−−−−
− −
−−−−−−−−−
−
,Ⓧ
.
, Contents
Prerequisites and Preliminaries 11
.7 The Axiom of Choice, Order and Zorn’s Lemma .............................. 11
7.1 Lattice ................................................................................... 11
7.2 Complete .............................................................................. 12
7.3 Well-ordering. ....................................................................... 13
7.4 Choice Function. ................................................................... 14
7.5 Semi-Lexicographic Order .................................................... 14
7.6 Projections ............................................................................ 15
7.7 Successors ........................................................................... 15
.8 Cardinal Numbers ............................................................................ 17
8.1 Pigeon-Hole Principle ........................................................... 17
8.2 Cardinality ............................................................................. 18
8.3 Countable ............................................................................. 19
8.4 Cardinal Arithmetic. .............................................................. 19
8.5 Cardinal Arithmetic Properties .............................................. 20
8.6 Finite Cardinal Arithmetic. .................................................... 21
8.7 Cardinal Order ...................................................................... 22
8.8 Countable Subsets ............................................................... 22
8.9 Cantor’s Diagonalization Method. ......................................... 23
8.10 Cardinal Exponents .............................................................. 23
8.11 Unions of Finite Sets............................................................. 25
8.12 Fixed Cardinal Unions .......................................................... 26
I Groups 27
I.1 Semigroups, Monoids, and Groups .................................................. 27
I.1.1 Non-group Objects................................................................ 27
I.1.2 Groups of Functions ............................................................. 28
I.1.3 Floops ................................................................................... 28
I.1.4 D4 Table............................................................................. 28
I.1.5 Order of Sn ..................................................................... 29
I.1.6 Klein Four Group .................................................................. 29
I.1.7 Z× p
............................................................................................................................................. 30
I.1.8 Q/Z – Rationals Modulo One ................................................ 30
I.1.9 Rational Subgroups .............................................................. 31
I.1.10 PruferGroup .......................................................................... 32
I.1.11 Abelian Relations .................................................................. 32
I.1.12 Cyclic Conjugates ................................................................. 33
I.1.13 Groups of Involutions ............................................................ 33
I.1.14 Involutions in Even Groups ................................................... 33
I.1.15 Cancellation in Finite Semigroups ........................................ 34
I.1.16 n-Product. ............................................................................. 35
I.2 Homomorphisms and Subgroups ..................................................... 36
I.2.1 Homomorphisms................................................................... 36
I.2.2 Abelian Automorphism. ........................................................ 37
I.2.3 Quaternions .......................................................................... 37
I.2.4 D4 in R2×2 .................................................................................................................. 38
3
Solutions Manual
Volume I: Introduction through Chapter IV
James Wilson
D4
⟨a 2, b⟩ ⟨a⟩ ⟨a ,2ab⟩
I
⟨b⟩ ⟨a2 b⟩ ⟨a2 ⟩ ⟨ab⟩ ⟨a 3b⟩
0
0 = C0(G) ≤ C1(G) ≤ ≤ Cn−1(G) ≤Cn(G) = G
II 0 = Gn ≤ Gn−1 ≤ ≤ G1 ≤ G0 = G
0 = Γn+1G≤ Γn G ≤ ≤ Γ2G ≤ Γ 1G = G
−
Commutative Local
Ring Ring
Field
−−−− −−− −
Unique
Integral
Factorization
Domain
III Ring Unital
Domain
Principal
Ring Ideal −−−−−−−−−
−
Domain
Skew
Field
Principal
Ideal Euclidean
Ring Domain
··········
Euclidean
Ring
= ≤
0 A B C 0
··········
IV
= ≤
0 A′ B′ C′ 0
··········
,−−−−− −−− −−−
−−−−−
−
−−−−−−−−−
−−−
−−−−−−−−−
− −
−−−−−−−−−
−
,Ⓧ
.
, Contents
Prerequisites and Preliminaries 11
.7 The Axiom of Choice, Order and Zorn’s Lemma .............................. 11
7.1 Lattice ................................................................................... 11
7.2 Complete .............................................................................. 12
7.3 Well-ordering. ....................................................................... 13
7.4 Choice Function. ................................................................... 14
7.5 Semi-Lexicographic Order .................................................... 14
7.6 Projections ............................................................................ 15
7.7 Successors ........................................................................... 15
.8 Cardinal Numbers ............................................................................ 17
8.1 Pigeon-Hole Principle ........................................................... 17
8.2 Cardinality ............................................................................. 18
8.3 Countable ............................................................................. 19
8.4 Cardinal Arithmetic. .............................................................. 19
8.5 Cardinal Arithmetic Properties .............................................. 20
8.6 Finite Cardinal Arithmetic. .................................................... 21
8.7 Cardinal Order ...................................................................... 22
8.8 Countable Subsets ............................................................... 22
8.9 Cantor’s Diagonalization Method. ......................................... 23
8.10 Cardinal Exponents .............................................................. 23
8.11 Unions of Finite Sets............................................................. 25
8.12 Fixed Cardinal Unions .......................................................... 26
I Groups 27
I.1 Semigroups, Monoids, and Groups .................................................. 27
I.1.1 Non-group Objects................................................................ 27
I.1.2 Groups of Functions ............................................................. 28
I.1.3 Floops ................................................................................... 28
I.1.4 D4 Table............................................................................. 28
I.1.5 Order of Sn ..................................................................... 29
I.1.6 Klein Four Group .................................................................. 29
I.1.7 Z× p
............................................................................................................................................. 30
I.1.8 Q/Z – Rationals Modulo One ................................................ 30
I.1.9 Rational Subgroups .............................................................. 31
I.1.10 PruferGroup .......................................................................... 32
I.1.11 Abelian Relations .................................................................. 32
I.1.12 Cyclic Conjugates ................................................................. 33
I.1.13 Groups of Involutions ............................................................ 33
I.1.14 Involutions in Even Groups ................................................... 33
I.1.15 Cancellation in Finite Semigroups ........................................ 34
I.1.16 n-Product. ............................................................................. 35
I.2 Homomorphisms and Subgroups ..................................................... 36
I.2.1 Homomorphisms................................................................... 36
I.2.2 Abelian Automorphism. ........................................................ 37
I.2.3 Quaternions .......................................................................... 37
I.2.4 D4 in R2×2 .................................................................................................................. 38
3
