Schedule
Term Fall 2026: 2026-08-24 through 2026-12-07.
| Unit | Topic focus |
|---|---|
| Week 1 — Sets, functions, and binary operations | sets and set operations as the working vocabulary, functions and the injective surjective bijective distinction, binary operations on a set, closure as a real condition, associativity and commutativity, identity elements and inverses, Cayley tables for finite operations, operations that fail an axiom |
| Week 2 — Permutations and the symmetric group | permutations as bijections of a finite set, composition of permutations and its order convention, cycle notation and disjoint cycle decomposition, the order of a permutation from its cycle type, transpositions and the parity of a permutation, the symmetric group on n letters, the alternating group |
| Week 3 — Divisibility and the division algorithm | the divisibility relation and its elementary properties, proving divisibility statements from the definition, the well ordering principle, the division algorithm with existence and uniqueness of quotient and remainder, common divisors, the greatest common divisor and its defining properties, linear combinations of two integers |
| Week 4 — The Euclidean algorithm, primes, and unique factorization | the Euclidean algorithm and why it terminates, Bezout’s identity by back substitution, relatively prime integers, primes and irreducibility in the integers, Euclid’s lemma, the fundamental theorem of arithmetic, the infinitude of the primes, least common multiple through the factorization |
| Week 5 — Equivalence relations, partitions, and congruence classes | relations on a set, reflexive symmetric and transitive properties, equivalence relations and their classes, the correspondence between equivalence relations and partitions, congruence modulo n as an equivalence relation, the congruence classes and the set of residues, well definedness of addition and multiplication on classes, the integers modulo n and their units |
| Week 6 — Groups and subgroups | the group axioms and what each one buys, first examples from numbers matrices and symmetries, uniqueness of the identity and of inverses, the cancellation laws, the order of a group, subgroups and the subgroup criterion, the centre and centralizers, subgroup lattices of small groups |
| Week 7 — Cyclic groups and the order of an element | the subgroup generated by a set, the order of an element and its relation to powers, cyclic groups and their generators, the classification of cyclic groups as the integers or the integers modulo n, subgroups of a cyclic group, the divisor lattice of a finite cyclic group, generators of the integers modulo n and the Euler totient count |
| Week 8 — Cosets and Lagrange’s theorem | left and right cosets of a subgroup, cosets as the classes of an equivalence relation, every coset has the size of the subgroup, the index of a subgroup, Lagrange’s theorem and its proof, the order of an element divides the order of the group, groups of prime order are cyclic, Fermat’s little theorem and Euler’s theorem as consequences, the converse of Lagrange fails |
| Week 9 — Direct products | the external direct product of two groups, the order and the element orders in a product, when a direct product of cyclic groups is cyclic, the Chinese remainder statement for the integers modulo a product, internal direct products and the recognition criterion, the Klein four group against the cyclic group of order four, building groups of a given order |
| Week 10 — Normal subgroups and quotient groups | conjugation and conjugate subgroups, normality and the equivalent tests for it, subgroups of index two are normal, the set of cosets as a group, well definedness of the quotient operation, the quotient group and its order, examples of quotients and what they forget, simple groups as those with no proper normal subgroup |
| Week 11 — Isomorphisms and structural invariants | the definition of an isomorphism of groups, what an isomorphism preserves and what it does not, exhibiting an explicit isomorphism, isomorphism as an equivalence relation on groups, structural invariants that separate groups including order abelianness element order profile and subgroup count, the classification of groups of small order, automorphisms of a group |
| Week 12 — Homomorphisms and the isomorphism theorems | group homomorphisms and their elementary properties, the kernel and the image as subgroups, the kernel is always normal and every normal subgroup is a kernel, the canonical projection onto a quotient, the first isomorphism theorem with proof, the correspondence between subgroups above the kernel, worked identifications of quotient groups |
| Week 13 — Cayley’s theorem and permutation representations | the left regular representation of a group, Cayley’s theorem and its proof, reading the representation off a Cayley table, the permutation representation of a small group written in cycle notation, the coset representation and the extended form of the theorem, what the theorem does and does not classify, the cost of the embedding |
| Week 14 — Rings and integral domains | the ring axioms and the two operations, commutative rings and rings with identity, first examples including the integers polynomials matrices and the integers modulo n, elementary consequences of the axioms, units and the group of units, zero divisors, integral domains, subrings and the subring test, ideals as the ring analogue of a normal subgroup |
| Week 15 — Fields, characteristic, and the arc of the course | the field axioms, fields as commutative rings in which every nonzero element is a unit, the integers modulo a prime as a field, a finite integral domain is a field, the characteristic of a ring and why it is zero or prime, the field of fractions of an integral domain, the containment of ring classes, the whole course seen through closure associativity identity and inverses |
The LMS remains authoritative for section logistics and graded details.