Algebraic Structures 1

Make the step from computing with numbers to reasoning about structure: take the arithmetic a student already trusts, notice that the properties doing the real work are closure, associativity, identity, and inverses, and then study every system that has them at once. The course builds the integers up carefully enough to see divisibility and congruence as the first examples, develops group theory to the point where isomorphism, quotients, and Cayley’s theorem say what a group really is, and ends by adding a second operation to reach rings, integral domains, and fields.

3 credits · in_person · Fall 2026

What you will learn

Decide whether a given rule on a set is a binary operation, and test it for closure, associativity, commutativity, an identity, and inverses.
Read and build a Cayley table, and use it to detect structural properties of a finite system.
Write a permutation in cycle notation, compose permutations, and decompose one into transpositions.
Determine the parity and the order of a permutation, and describe the symmetric and alternating groups on a finite set.
Apply the division algorithm, and prove basic divisibility statements directly from the definition.
Compute a greatest common divisor by the Euclidean algorithm and express it as an integer combination by back-substitution.
Prove and use Euclid’s lemma, and state the fundamental theorem of arithmetic together with what uniqueness means in it.
Verify that a relation is an equivalence relation and describe the partition its classes induce.
Compute in the integers modulo n, decide when a congruence class is invertible, and solve linear congruences.
State the group axioms, verify or refute that a given set with a given operation is a group, and derive the elementary consequences of the axioms.
Apply the subgroup criterion, and identify the centre, a centralizer, and the subgroup generated by a set.
Determine the order of an element, describe every subgroup of a cyclic group, and decide when a group is cyclic.
Partition a group into cosets of a subgroup, prove Lagrange’s theorem, and draw its standard consequences for finite groups.
Construct external and internal direct products, and decide when a group decomposes as a direct product of two subgroups.
Test a subgroup for normality, construct the quotient group, and compute in it.
Decide whether two groups are isomorphic, exhibit an explicit isomorphism, or separate them by a structural invariant.
Verify that a map is a homomorphism, compute its kernel and image, and apply the first isomorphism theorem.
State and prove Cayley’s theorem, and construct the permutation representation of a small finite group.
Verify the ring axioms, identify units and zero divisors, and distinguish a ring from an integral domain and from a field.
Determine the characteristic of a ring, prove that a finite integral domain is a field, and recognise when the integers modulo n form a field.

How the course runs

Algebraic Structures 1 meets as Monday and Wednesday, 1:40 to 2:55 in the afternoon, in person through Fall 2026, 2026-08-24 to 2026-12-07, across 15 units. This site is a draft. Zero student cost: every reading it points to is openly available online at no charge, including the textbook the syllabus requires, and the notes are written to stand on their own. The course site is the primary place to read the material; the learning management system remains the authoritative source for section logistics, the course calendar, quiz and examination dates and coverage, and every graded detail. The syllabus refers to a course calendar that was not supplied with it, so no date, coverage, or assessment logistic appears anywhere on this site.

Start here

Begin with Week 1 — Sets, functions, and binary operations, then follow the unit list in the sidebar.

How to use this site

Everything you need is on this site, at no cost: unit notes, the syllabus summary, the schedule, and supporting resources. The LMS remains authoritative for section logistics and graded details.