UMD 403: Undergraduate Algebra
1
Preliminaries
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1.1
Overview
1.2
Notation and Well-Ordered Property
1.3
Existence
1.4
Uniqueness
1.5
Euclid’s Lemma and the more standard uniqueness proof
1.6
Attempt at getting the history right
1.7
The division algorithm with remainder
1.8
Greatest common divisor and least common multiple
2
Binary operations, monoids and groups
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2.1
Binary Operations
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2.1.1
Multiplication Tables
Exercises
2.2
Monoids and Groups
2.3
Submagmas, submonoids and subgroups
2.4
The monoid of subsets of a monoid
2.5
Subgroup generated by a set
2.6
Powers of elements
2.7
Subgroups of cyclic groups
2.8
Cyclic Groups and Orders of Elements
2.9
Centralizers and Center
2.10
Partitions of sets and equivalence relations
2.11
Cosets and Lagrange’s Theorem
2.12
Order of product of subgroups
3
Basic group theory
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3.1
Homomorphisms
3.2
Examples of Homomorphisms
3.3
The set of homomorphisms
3.4
Images and preimages
3.5
Images, inverse images and kernels
3.6
Normal subgroups
3.7
Quotients by normal subgroups
3.4 Images and preimages