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Syllabus Math 4083/5083 – 001 – Fall 2025

Blake Farman

Section 6 Schedule

Subsection 6.1 September 4 – September 5

September 5
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Binary Operations
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Subsection 6.2 September 8 – September 12

September 8
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Groups
  • Axioms,
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  • Elementary Properties
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September 10
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September 12
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Abelian Examples
  • Integers Modulo \(n\)
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Subsection 6.3 September 15 – September 19

September 15
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Abelian Examples
  • Roots of Unity
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September 17
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Nonabelian Examples
  • Symmetric Groups
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Subgroups
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September 19
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Subgroups
  • Images and Preimages
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  • Kernels
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  • Subgroup Criterion
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Cyclic Groups
  • Integer Powers of Group Elements
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  • Cyclic Groups and Subgroups
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  • Basic Examples
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Subsection 6.4 September 22 – September 26

September 22
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Cyclic Groups
  • Classification of Cyclic Groups
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September 24
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Cyclic Groups
  • Subgroups of Cyclic Groups
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September 26
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Groups of Permutations
  • Cayley’s Theorem
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Subsection 6.5 September 29 – October 3

September 29
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October 1
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Review
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October 3
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Exam 1
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Subsection 6.6 October 6 – October 10

October 6
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Cosets and the Theorem of Lagrange
  • Left/Right Equivalence Modulo a Subgroup
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  • Left and Right Cosets
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  • Lagrange’s Theorem
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  • Index of a Subgroup
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  • Left/Right Cosets and Kernels
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October 8
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Factor Groups
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October 10
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Factor Groups
  • Universal Mapping Property for Quotients
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  • First Isomorphism Theorem for Groups
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Factor-Group Computations and Simple Groups
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Subsection 6.7 October 13 – October 17

October 13
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Rings and Fields
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October 15
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Rings and Fields
  • Basic Examples
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  • Basic Properties of Rings
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October 17
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Rings and Fields
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Subsection 6.8 October 20– October 24

October 20
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Rings and Fields
  1. Fields
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  2. Subrings and Subfields
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October 22
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Review
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October 24
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Exam 2
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Subsection 6.9 October 27 – October 31

October 27
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Integral Domains
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October 29
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Rings of Polynomials
  • Roots of Polynomials
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  • Evaluation Homomorphism for Fields
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October 31
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Class Cancelled by Provost
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Subsection 6.10 November 3– November 7

November 3
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Homomorphisms and Factor Rings
  • Ideals
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  • Quotient Rings
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  • Universal Mapping Property for Quotients (Reprise)
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  • First Isomorphism Theorem for Rings
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November 5
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Prime and Maximal Ideals
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November 7
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Review
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Subsection 6.11 November 10 – November 13

November 10
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Review
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November 12
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Final Exam
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Warning 6.1.

The instructor reserves the right to modify the schedule as needed.
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