Lecture Notes for Abstract Algebra I
James S. Cook
Liberty University
Department of Mathematics
Fall 2016
,2
preface
Abstract algebra is a relatively modern topic in mathematics. In fact, when I took this course it
was called Modern Algebra. I used the fourth ed. of Contemporary Abstract Algebra by Joseph
Gallian. It happened that my double major in Physics kept me away from the lecture time for the
course. I learned this subject first from reading Gallian’s text. In my experience, it was an excellent
and efficient method to initiate the study of abstract algebra. Now, the point of this story is not
that I want you to skip class and just read Gallian. I will emphasize things in a rather different
way, but, certainly reading Gallian gives you a second and lucid narrative to gather your thoughts
on this fascinating topic. I provide these notes to gather ideas from Gallian and to add my own.
sources
I should confess, I have borrowed many ideas from:
1. Contemporary Abstract Algebra by Joseph Gallian
2. the excellent lectures given by Professor Gross of Harvard based loosely on Artin’s Algebra
3. Dummit and Foote’s Abstract Algebra
4. Fraleigh
5. Rotman
style guide
I use a few standard conventions throughout these notes. They were prepared with LATEX which
automatically numbers sections and the hyperref package provides links within the pdf copy from
the Table of Contents as well as other references made within the body of the text.
I use color and some boxes to set apart some points for convenient reference. In particular,
1. definitions are in green.
2. remarks are in red.
3. theorems, propositions, lemmas and corollaries are in blue.
4. proofs start with a Proof: and are concluded with a .
However, I do make some definitions within the body of the text. As a rule, I try to put what I
am defining in bold. Doubtless, I have failed to live up to my legalism somewhere. If you keep a
list of these transgressions to give me at the end of the course it would be worthwhile for all involved.
The symbol indicates that a proof is complete. The symbol O indicates part of a proof is done,
but it continues.
As I add sections, the Table of Contents will get longer and eventually change the page numbering
of the later content in terms of the pdf. When I refer to page number, it will be the document
numbering, not the pdf numbering.
,Contents
1 Group Theory 1
1.1 Lecture 1: an origin story: groups, rings and fields . . . . . . . . . . . . . . . . . . . 1
1.2 Lecture 2: on orders and subgroups . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8
1.3 Lecture 3: on the dihedral group and symmetries . . . . . . . . . . . . . . . . . . . . 13
1.4 Lecture 4: back to Z-number theory . . . . . . . . . . . . . . . . . . . . . . . . . . . 18
1.4.1 Z-Basics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18
1.4.2 division algorithm . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19
1.4.3 divisibility in Z . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21
1.5 Lecture 5: modular arithmetic and groups . . . . . . . . . . . . . . . . . . . . . . . . 27
1.6 Lecture 6: cyclic groups . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40
1.7 Lecture 7: classification of cyclic subgroups . . . . . . . . . . . . . . . . . . . . . . . 46
1.8 Lecture 8: permutations and cycle notation . . . . . . . . . . . . . . . . . . . . . . . 50
1.9 Lecture 9: theory of permutations . . . . . . . . . . . . . . . . . . . . . . . . . . . . 54
2 On the Structure of Groups 61
2.1 Lecture 10: homomorphism and isomorphism . . . . . . . . . . . . . . . . . . . . . . 62
2.2 Lecture 11: isomorphism preserves structure . . . . . . . . . . . . . . . . . . . . . . . 67
2.3 Lecture 12: cosets and Lagrange’s Theorem . . . . . . . . . . . . . . . . . . . . . . . 73
2.4 Lecture 13: on dividing and multiplying groups . . . . . . . . . . . . . . . . . . . . . 78
2.5 Lecture 14: on the first isomorphism theorem . . . . . . . . . . . . . . . . . . . . . . 83
2.5.1 classification of groups up to order 7 . . . . . . . . . . . . . . . . . . . . . . . 83
2.5.2 a discussion of normal subgroups . . . . . . . . . . . . . . . . . . . . . . . . . 84
2.5.3 first isomorphism theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 86
2.6 Lecture 15: first isomorphism theorem, again! . . . . . . . . . . . . . . . . . . . . . . 89
2.7 Lecture 16: direct products inside and outside . . . . . . . . . . . . . . . . . . . . . . 93
2.7.1 classification of finite abelian groups . . . . . . . . . . . . . . . . . . . . . . . 97
2.8 Lecture 17: a little number theory and encryption . . . . . . . . . . . . . . . . . . . 99
2.8.1 encryption . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 102
2.9 Lecture 18: group actions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 106
2.10 Lecture 19: orbit stabilizer theorem and conjugacy . . . . . . . . . . . . . . . . . . . 110
2.11 Lecture 20: Cauchy and Sylow theorems . . . . . . . . . . . . . . . . . . . . . . . . . 113
2.12 Lecture 21: lattice theorem, finite simple groups . . . . . . . . . . . . . . . . . . . . 116
2.13 Lecture 22: Boolean group, rank nullity theorem . . . . . . . . . . . . . . . . . . . . 116
3 Introduction to Rings and Fields 117
3.1 Lecture 23: rings and integral domains . . . . . . . . . . . . . . . . . . . . . . . . . . 118
3.2 Lecture 24: ideals and factor rings . . . . . . . . . . . . . . . . . . . . . . . . . . . . 126
3
, 4 CONTENTS
3.3 Lecture 25: ring homomorphism . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 134
3.4 Lecture 26: polynomials in an indeterminant . . . . . . . . . . . . . . . . . . . . . . 140
3.5 Lecture 27: factorization of polynomials . . . . . . . . . . . . . . . . . . . . . . . . . 146
3.6 Lecture 28: divisibility in integral domains I . . . . . . . . . . . . . . . . . . . . . . . 153
3.7 Lecture 29: divisibility in integral domains II . . . . . . . . . . . . . . . . . . . . . . 157
3.8 Lecture 30: extension fields . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 164
3.9 Lecture 31: algebraic extensions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 170
3.10 Lecture 32: algebraically closed fields . . . . . . . . . . . . . . . . . . . . . . . . . . . 173
James S. Cook
Liberty University
Department of Mathematics
Fall 2016
,2
preface
Abstract algebra is a relatively modern topic in mathematics. In fact, when I took this course it
was called Modern Algebra. I used the fourth ed. of Contemporary Abstract Algebra by Joseph
Gallian. It happened that my double major in Physics kept me away from the lecture time for the
course. I learned this subject first from reading Gallian’s text. In my experience, it was an excellent
and efficient method to initiate the study of abstract algebra. Now, the point of this story is not
that I want you to skip class and just read Gallian. I will emphasize things in a rather different
way, but, certainly reading Gallian gives you a second and lucid narrative to gather your thoughts
on this fascinating topic. I provide these notes to gather ideas from Gallian and to add my own.
sources
I should confess, I have borrowed many ideas from:
1. Contemporary Abstract Algebra by Joseph Gallian
2. the excellent lectures given by Professor Gross of Harvard based loosely on Artin’s Algebra
3. Dummit and Foote’s Abstract Algebra
4. Fraleigh
5. Rotman
style guide
I use a few standard conventions throughout these notes. They were prepared with LATEX which
automatically numbers sections and the hyperref package provides links within the pdf copy from
the Table of Contents as well as other references made within the body of the text.
I use color and some boxes to set apart some points for convenient reference. In particular,
1. definitions are in green.
2. remarks are in red.
3. theorems, propositions, lemmas and corollaries are in blue.
4. proofs start with a Proof: and are concluded with a .
However, I do make some definitions within the body of the text. As a rule, I try to put what I
am defining in bold. Doubtless, I have failed to live up to my legalism somewhere. If you keep a
list of these transgressions to give me at the end of the course it would be worthwhile for all involved.
The symbol indicates that a proof is complete. The symbol O indicates part of a proof is done,
but it continues.
As I add sections, the Table of Contents will get longer and eventually change the page numbering
of the later content in terms of the pdf. When I refer to page number, it will be the document
numbering, not the pdf numbering.
,Contents
1 Group Theory 1
1.1 Lecture 1: an origin story: groups, rings and fields . . . . . . . . . . . . . . . . . . . 1
1.2 Lecture 2: on orders and subgroups . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8
1.3 Lecture 3: on the dihedral group and symmetries . . . . . . . . . . . . . . . . . . . . 13
1.4 Lecture 4: back to Z-number theory . . . . . . . . . . . . . . . . . . . . . . . . . . . 18
1.4.1 Z-Basics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18
1.4.2 division algorithm . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19
1.4.3 divisibility in Z . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21
1.5 Lecture 5: modular arithmetic and groups . . . . . . . . . . . . . . . . . . . . . . . . 27
1.6 Lecture 6: cyclic groups . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40
1.7 Lecture 7: classification of cyclic subgroups . . . . . . . . . . . . . . . . . . . . . . . 46
1.8 Lecture 8: permutations and cycle notation . . . . . . . . . . . . . . . . . . . . . . . 50
1.9 Lecture 9: theory of permutations . . . . . . . . . . . . . . . . . . . . . . . . . . . . 54
2 On the Structure of Groups 61
2.1 Lecture 10: homomorphism and isomorphism . . . . . . . . . . . . . . . . . . . . . . 62
2.2 Lecture 11: isomorphism preserves structure . . . . . . . . . . . . . . . . . . . . . . . 67
2.3 Lecture 12: cosets and Lagrange’s Theorem . . . . . . . . . . . . . . . . . . . . . . . 73
2.4 Lecture 13: on dividing and multiplying groups . . . . . . . . . . . . . . . . . . . . . 78
2.5 Lecture 14: on the first isomorphism theorem . . . . . . . . . . . . . . . . . . . . . . 83
2.5.1 classification of groups up to order 7 . . . . . . . . . . . . . . . . . . . . . . . 83
2.5.2 a discussion of normal subgroups . . . . . . . . . . . . . . . . . . . . . . . . . 84
2.5.3 first isomorphism theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 86
2.6 Lecture 15: first isomorphism theorem, again! . . . . . . . . . . . . . . . . . . . . . . 89
2.7 Lecture 16: direct products inside and outside . . . . . . . . . . . . . . . . . . . . . . 93
2.7.1 classification of finite abelian groups . . . . . . . . . . . . . . . . . . . . . . . 97
2.8 Lecture 17: a little number theory and encryption . . . . . . . . . . . . . . . . . . . 99
2.8.1 encryption . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 102
2.9 Lecture 18: group actions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 106
2.10 Lecture 19: orbit stabilizer theorem and conjugacy . . . . . . . . . . . . . . . . . . . 110
2.11 Lecture 20: Cauchy and Sylow theorems . . . . . . . . . . . . . . . . . . . . . . . . . 113
2.12 Lecture 21: lattice theorem, finite simple groups . . . . . . . . . . . . . . . . . . . . 116
2.13 Lecture 22: Boolean group, rank nullity theorem . . . . . . . . . . . . . . . . . . . . 116
3 Introduction to Rings and Fields 117
3.1 Lecture 23: rings and integral domains . . . . . . . . . . . . . . . . . . . . . . . . . . 118
3.2 Lecture 24: ideals and factor rings . . . . . . . . . . . . . . . . . . . . . . . . . . . . 126
3
, 4 CONTENTS
3.3 Lecture 25: ring homomorphism . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 134
3.4 Lecture 26: polynomials in an indeterminant . . . . . . . . . . . . . . . . . . . . . . 140
3.5 Lecture 27: factorization of polynomials . . . . . . . . . . . . . . . . . . . . . . . . . 146
3.6 Lecture 28: divisibility in integral domains I . . . . . . . . . . . . . . . . . . . . . . . 153
3.7 Lecture 29: divisibility in integral domains II . . . . . . . . . . . . . . . . . . . . . . 157
3.8 Lecture 30: extension fields . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 164
3.9 Lecture 31: algebraic extensions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 170
3.10 Lecture 32: algebraically closed fields . . . . . . . . . . . . . . . . . . . . . . . . . . . 173
