Groups
Basic Definitions and Examples
• Group; Abelian group; order of a group; order of an element of a group
• The dihedral group D_2n of order 2n; the symmetric group S_n; the matrix groups GL_n(F); the quaternion group Q₈
Subgroups, Homomorphisms and Quotient Groups
• Group homomorphisms and isomorphisms; kernels and images; group actions
• Subgroups, normal subgroups, lattice of subgroups
• Quotient groups; left (or right) cosets; Lagrange's Theorem; Cauchy's Theorem; The First Isomorphism Theorem; The Second Isomorphism Theorem; The Third Isomorphism Theorem
Group Actions and Representations
• Group action on a set; permutation representation given by an action; stabilizers; orbits; transitive actions; action by left multiplication; Cayley's Theorem; action by conjugation; The Class Equation; conjugacy classes in S_n
Abelian Groups
• Direct products of abelian groups
• The Fundamental Theorem of Finitely Generated Abelian Groups

Rings
Basic Definitions and Examples
• Rings; division rings; fields; zero divisors; Z, Q, R, and C; Z_n (integers modulo n); H (quaternions); ; matrix rings; R[x]
Ring Homomorphisms, Ideals and Quotients
• Ring homomorphisms, kernels and images; ideals; quotient rings; The First Isomorphism Theorem; principal ideals; maximal ideals; prime ideals
Polynomial Rings, Factorization and Irreducibility
• Polynomial rings R[x], R[x₁,x₂, ... , x_n]; the division algorithm and the Euclidean algorithm; uniqueness of factorization, (principal) ideals in F[x]; irreducibility; Gauss' Lemma; Eisenstein's Criterion; maximal ideals in F[x], (F is a field)

Fields
Field Extensions
• Characteristic; prime subfield; extension field; degree of an extension; basis for F[x]/(p(x)); simple extension; algebraic extensions; minimal polynomial
• Splitting fields and Galois groups
• Finite fields
• Fundamental theorem of Galois theory
• Galois group of a cubic over Q

References:

Abstract Algebra, 3rd Edition, Dummit and Foote.