You are responsible for material from the entire semester, although there will be an emphasis on groups. Here is a brief list of topics we've covered; numbers in parentheses refer to sections of the text by Judson.

• Foundational material Sets, functions, inverse mappings, equivalence relations, partitions. (1.2)
• Induction Inductive proofs, principle of well-ordering. (2.1)
• Integers Division algorithm, gcd's, Euclidean algorithm, primes, unique factorization. (2.2)
• Rings Definitions and examples, basic properties of 0, 1 and -a, zero divisors, units, subrings, integral domains, fields. (16.1, 16.2)
• Polynomial rings Basic definitions and properties, division algorithm, gcd. (17.1, 17.2)
• Homomorphisms and quotients Homomorphisms, kernels, ideals, quotient rings, Chinese remainder theorem. (16.3, 16.5)
• Groups Basic definitions and properties, examples, subgroups, orders and exponents. (3.1, 3.2, 3.3)
• Cyclic groups Cyclic groups, subgroups, generators. (4.1)
• Permutation Notation, disjoint cycle notation, transpositions, even/odd. (5.1, some of 5.2)
• Cosets Definitions, Lagrange's theorem, Fermat's little theorem. (6.1, 6.2, 6.3)
• Homomorphisms and quotients Homomorphisms, kernels, normal subgroups, quotient groups. (11.1, 10.1, some of 12.2)