Abstract Algebra Tuition in Singapore

Abstract Algebra

In abstract algebra, I introduce axiomatic structures that are algebraic in nature, such as groups, rings, fields and modules.

Groups, subgroups, cyclic groups, cosets, Lagrange's theorem, groups of permutations
Group actions, orbits, stabilisers, normal subgroups, quotient groups, the isomorphism theorems, Sylow's Theorem
Direct and semidirect products, automorphism groups, nilpotent groups, solvable groups
Rings, ideals, quotient rings, integral domains, unique factorization domains, principal ideal domains, rings of fractions, polynomial rings over fields
Modules over a ring, submodules, quotient modules, module homomorphisms, modules over polynomial rings, modules over principal ideal domains
Fields, algebraic closure, Galois Theory

Where abstract algebra usually trips students up

Abstract algebra is many students' first fully axiomatic course, and the difficulty is the style of thinking as much as the content.

The leap into abstraction. The subject starts from axioms and only later shows you the examples, which is the reverse of how most earlier maths was taught.
Cosets and quotient groups. This is the single biggest hurdle for most students, and rushing it makes everything built on top feel shaky.
Why normality matters. It is easy to learn the definition of a normal subgroup without seeing that it is exactly what lets a quotient group exist.
Building intuition from examples. The theory only comes alive once you carry a handful of concrete groups and rings in your head to test claims against.
Reading the isomorphism theorems as machinery. They feel mechanical until you see them as the same idea recurring, at which point the proofs shorten dramatically.
Writing structural proofs. Marks come from clean arguments about arbitrary elements, not from computation, and that takes deliberate practice.

Before you start: an abstract algebra prerequisites checklist

Abstract algebra needs very little calculus, but it leans heavily on comfort with proof and a few foundations. Before week one, run through the list below and note anything that feels shaky.

What to be comfortable with before week one
Be comfortable with	Why it matters for abstract algebra
Proof technique	The course is almost entirely proof, so fluency with direct, contradiction and contrapositive arguments is the real prerequisite.
Sets, functions and relations	Injections, surjections and especially equivalence relations underpin cosets and quotient structures.
Equivalence relations	Partitioning a set by an equivalence relation is exactly the mechanism behind quotient groups and rings.
Elementary number theory	Divisibility and modular arithmetic give the first and friendliest examples of groups and rings.
Basic linear algebra	Vector spaces are a familiar example to lean on when modules are introduced.
Mathematical notation	Reading dense symbolic statements accurately matters more here than raw calculation.

A one-semester self-study plan for independent learners

Abstract algebra is a wonderful subject to study independently, provided you work through plenty of examples rather than only reading. If you are teaching yourself, here is a thirteen-week route map built around texts pitched at different levels. Move at a pace that suits you, and get in touch if cosets and quotient groups are not yet settling, since that is where a few sessions pay off most.

Texts I often point self-study students to
Text	Author	Best for
Contemporary Abstract Algebra	Joseph Gallian	A friendly, example-rich first course that builds intuition gently
Abstract Algebra	Dummit & Foote	The comprehensive standard reference for the whole subject
Algebra	Michael Artin	A more geometric flavour, strong on the link with linear algebra
Topics in Algebra	I. N. Herstein	A concise classic that rewards careful reading
Galois Theory	Ian Stewart	An accessible route into the field-theory material at the end

A thirteen-week plan
Week	Focus	Suggested reading	Checkpoint
1	Groups, subgroups and examples	Gallian or Dummit & Foote, opening chapters	Recognise a group and verify the axioms confidently
2	Cyclic groups and order	Gallian	Work with generators and the order of elements
3	Permutation groups	Gallian or Dummit & Foote	Compute in the symmetric and alternating groups
4	Cosets and Lagrange's theorem	Gallian	Use cosets and prove Lagrange's theorem
5	Normal subgroups and quotient groups	Dummit & Foote	Build a quotient group and explain why normality is needed
6	Homomorphisms and the isomorphism theorems	Gallian or Dummit & Foote	Apply the first isomorphism theorem fluently
7	Group actions and the orbit-stabiliser theorem	Dummit & Foote	Use a group action to count or to classify
8	The Sylow theorems	Dummit & Foote	Apply the Sylow theorems to analyse a finite group
9	Rings, ideals and quotient rings	Gallian or Dummit & Foote	Form a quotient ring and identify its ideals
10	Integral domains, PIDs and UFDs	Dummit & Foote	Place a given ring in the right part of the hierarchy
11	Polynomial rings and factorisation	Dummit & Foote	Test irreducibility and factor in polynomial rings
12	Fields and field extensions	Stewart or Dummit & Foote	Compute the degree of a field extension
13	Introduction to Galois theory and review	Stewart	See how the Galois correspondence ties the course together

Frequently Asked Questions (FAQ) - Abstract Algebra Tuition

Which abstract algebra topics and modules do you cover?

Group theory from the axioms through to the Sylow theorems, ring and ideal theory, modules including the structure theorem over a principal ideal domain, and field theory up to Galois theory. That covers the abstract-algebra modules maths majors take at NUS, NTU and the other local universities, up to honours level. Send me your syllabus and I will let you know how well it fits.

I keep getting lost at cosets and quotient groups. Can you help?

Yes, and this is by far the most common sticking point, so you are in good company. We slow right down here, build the idea from equivalence relations and small concrete examples, and only move on once quotients feel natural. Almost everything later in the course depends on getting this one idea solid.

How do I get better at writing the proofs this course expects?

By arguing about arbitrary elements rather than computing with specific ones, and by practising with feedback. We work on stating exactly what is assumed, what is to be shown, and which definition or theorem bridges the two. Once you see the same handful of proof shapes recurring, the writing becomes much steadier.

Do you cover Galois theory and module theory as well?

Yes. Field extensions and Galois theory, and modules over a ring including the structure theorem over a principal ideal domain, are all part of the work I teach, and they sit on this same page rather than a separate one.

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