Math 451 - Abstract Algebra I

3 credits. Offered in the Fall semester. Official course information. Prerequisites: special permission.

Description

Math 451 is the first course in the honors abstract algebra sequence, 451/452. Abstract algebra studies operations defined on arbitrary sets (e.g. how to multiply functions), their properties, and how these compare with the properties of usual operations defined on numbers (e.g. associativity, commutativity, etc.). There are three main abstract algebraic structures studied in an undergraduate curriculum: groups, rings, and fields. Math 451 studies groups while 452 will study rings and fields.

Math 351 is another abstract algebra course offered by the mathematics department, but it differs substantially from the 451-452 sequence. Math 351 covers much of the material from the 451-452 sequence in one semester, but in less depth.

Topics covered

Topics usually covered in Math 451 include:

  • Definition of binary operations and groups.
  • Examples of groups: symmetric groups, dihedral groups, integers modulo n and cyclic groups, etc.
  • Subgroups and group homomorphisms/isomorphisms
  • Direct products of groups
  • Cosets and Lagrange’s theorem
  • Conjugation, normal subgroups, and quotient groups
  • Group actions and the orbit-stabilizer theorem
  • Cayley’s theorem
  • Conjugacy in groups: conjugacy classes, centralizers, and the class equation
  • Sylow theorems and their applications
  • Representation theory of finite groups

Note that the exact list of topics may vary by instructor and semester. Additional topics may include free groups and presentations and semi-direct products.

Comments on prerequisites

Unofficial/suggested additional prerequisites: Math 350.

Though Math 350 is not an official prerequisite, successful completion of Math 350 is typically needed in order to be granted permission to take 451. The honors section of Math 350 is strongly preferred as well. Math 351 can also serve as good preparation, but the significant overlap of content between 351 and the 451/452 sequences makes this option redundant.

Resources

Textbook

The usual textbook for the class Algebra (2nd edition) by Michael Artin.

Additional materials

Notes taken from a comparable course to Math 451 at MIT are available online here. Similar notes covering representation theory of finite groups may be found on the page for Math 452.

Other useful textbooks include Abstract Algebra by David Dummit and Richard Foote, a more advanced textbook typically used in graduate algebra courses but has similar topic coverage.

Some past course webpages are available online.