• Direct products, Normal subgroups, Quotient groups, and the isomorphism theorems.
• Groups acting on sets; orbits and stabilizers. Applications: class formula, centralizers and normalizers, centers of finite p-groups. Conjugacy classes of S_n
• Sylow's Theorems, Solvable groups, Simple groups, Examples: Finite groups of small order (<=8).
• Structure of finitely generated abelian groups.

References: Lang, Chapter I; Dummit and Foote, Part I; Rotman; Artin, Chapter 12, Section 4

• Vector spaces, Linear dependence/independence, Bases, Matrices and linear maps.
• Eigenspaces and eigenvectors, characteristic polynomial.
• Jordan normal form. Computation of matrix-valued functions (Aⁿ, e^A etc).

References: Lang, Chapters XIII and XIV; Dummit and Foote, Chapter 11, Artin Chapter 4, Sections 6--8

• Rings, subrings, fields, ideals, homomorphisms, isomorphism theorems, polynomial rings, rings of formal power series.
• Integral domains, Euclidean domains, PID's. UFD's and Gauss's Lemma ( F[x₁,..., x_n] is an UFD). Quadratic integer rings.
• Prime ideals, maximal ideals. The Chinese remainder Theorem. Fields of fractions.

References: Lang, Chapters II, III, V, and VI; Jacobson, Chapter 2; Dummit and Foote, Part II.