Problem types
| Problem type | Page |
|---|---|
| Show that \(G\) is not simple, or classify the groups of order \(n\) | Show $G$ is not simple |
| Decide whether a group is abelian | Is this group abelian? |
| Place a ring among the classes of rings, or decide whether every ring of one class belongs to another | Which kind of ring is this? |
| Classify a finitely generated module or abelian group | Classify this module |
| Find the canonical form of a matrix | Find the canonical form |
| Compute a Galois group | Compute this Galois group |
The structure theorem for finitely generated modules over a principal ideal domain (Modules over a PID) specializes over \(\ZZ\) to the classification of finitely generated abelian groups and over \(k[x]\) to the rational canonical form of a matrix over a field \(k\).
Chapters
Groups – subgroups, quotients, the standard families.
Group actions – Lagrange, orbit-stabilizer, the class equation, Sylow.
Rings and ideals – the classes of rings, ideals, factorization.
Modules – the structure theorem, free and projective modules.
Linear algebra – the canonical forms.
Fields – extensions, splitting, separability, finite fields.
Galois theory – the correspondence, and computing Galois groups.
Representations – Maschke, Schur, characters.
Reference pages:
Review sheet – the main theorems, grouped by chapter.
Counterexamples – grouped by the statement each refutes.
Groups of small order – the groups of orders \(p\), \(p^2\), and \(pq\), and of each composite order up to \(20\).
Also: resources and workshops.
Syllabus
See Syllabus.
Acknowledgements
Thanks to Paco Adajar and Swaroop Hegde for solutions and proofs. Many solutions contain input from other graduate students and faculty at UGA, and from questions answered on Math Stack Exchange and MathOverflow.
Extra drill problems are drawn from Pierce’s algebra qual collections at UC Riverside, on groups and on Galois theory.