Math 413 syllabus

Abstract Algebra I (W)

Course Description: An introduction to group theory and ring theory. Topics include permutations and symmetries, subgroups, quotient groups, homomorphisms, as well as examples of rings, integral domains, and fields.
Prerequisites: MA 237 Minimum Grade of C and (MA 311 Minimum Grade of C or MA 320 Minimum Grade of C or MA 334 Minimum Grade of C) and (EH 102 Minimum Grade of C or EH 105 Minimum Grade of C).

Suggested Text: A Book of Abstract Algebra (Second Edition) by Charles C. Pinter, Dover Publications, Inc.
Coverage: Selection of topics from chapters 1 – 5 and chapters 7 – 20.


Learning outcomes: Upon the successful completion of the course a student will:

  • write short proofs (direct, by contradiction, and using the contrapositive)
  • disprove algebraic statements by finding a counterexample
  • state, justify, and apply basic properties of groups, rings, and fields
  • verify that a given subset of a group is a (normal) subgroup
  • verify that a given function is a homomorphism (isomorphism)
  • state and prove Cayley's theorem
  • find the order of a given group element
  • state and prove Cauchy's theorem (if time permits)
  • state and prove the structure theorem for cyclic groups
  • state and prove that subgroups of cyclic groups are cyclic
  • find all the cosets of a subgroup in a group
  • state and prove Lagrange's theorem
  • determine all groups of a given order ≤ 10 up to isomorphism
  • verify that a given subset of a ring is a subring (ideal)
  • state and prove the fundamental homomorphism theorems for groups and rings
  • state and prove the isomorphism theorems for groups and rings
  • state and prove the correspondence theorems for (normal) subgroups and subrings (ideals) (if time permits)
  • prove that two groups or rings are isomorphic or are not isomorphic
  • find the characteristic of an integral domain
  • construct the field of quotients of an integral domain