# 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