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