MAS 5311 Introduction to Algebra 1

The text for the course is Dummit and Foote, but the topics should be essentially the same whichever text, if any, is used.

This exam is on Group theory. The exam will test for:

  • Ability to write clear, rigorous proofs.
  • A solid understanding of the basic definitions.
  • Thorough understanding of the statements and proofs of theorems.
  • Ability to apply theorems to specific situations.
  • Familiarity with a broad range of examples and ability to do computations with them.

All topics are examinable except for those explicitly excluded. In the case of very long proofs, exam questions may involve only a part of the argument. In exams it is generally allowed to apply main theorems by quoting them correctly, as long as doing so does not result in circular logic, as would be the case if the question is really asking for part of the proof of the theorem quoted.

The modules below (in order) correspond to the first six chapters of Dummit and Foote.

  1. The section on generators and relations is not examinable. Topics: Definition of groups, basic examples, dihedral groups, symmetric groups, matrix groups, the quaternion group, homomorphisms and isomorphisms, group actions.
  2. Subgroup lattices of particular groups should be studied but not memorized Topics: Subgroups, centralizers and normalizers, cyclic groups, subgroup generated by a subset, the subgroup lattice.
  3. Topics: Cosets, normal subgroups, quotient groups, cosets and Lagrange’s Theorem, isomorphism theorems, simple groups, composition series, Jordan Hoelder theorem, Alternating groups.
  4. Topics: Permutation representations, actions on coset spaces, action by conjugation, automorphisms, Sylow theorems, simplicity of alternating groups.
  5. The lists of small groups in 5.3 should be studied, but not memorized. Topics: Direct products, the fundamental theorem for finite abelian groups (statement and applications are examinable, but not the proof), semidirect products, examples of groups of small order.
  6. Sections 6.2 and 6.3 are optional and not examinable. Topics: p-groups, nilpotent groups, solvable groups; Not for Examination: Free groups, generators and relations

Two sets of practice problems are available: