MH2220: pass the exams, not just read the notes
Your complete guide to Nanyang Technological University's algebra i course. See where the marks are, work real practice questions, and study with an AI tutor that knows MH2220.
Sia generates MH2220 practice questions, walks through group axioms and properties of groups step by step, and quizzes you on the material the exam weights most heavily.
Sharpen your argument
You must show a subgroup H of G is normal. Which approach is correct, and what does normality actually buy you?
State the definition being tested. H is normal in G when conjugation by any element of G maps H back to itself: gHg-inverse equals H for all g. Equivalently, left and right cosets coincide.
Name what you gain. Normality is precisely what makes the quotient group G/H exist — that is week 7 of the published schedule, and weeks 8 and 9 then spend two full weeks on the Isomorphism Theorems, which relate homomorphisms, kernels and quotients.
Dispose of the distractors. Option B confuses the subgroup being abelian with the group being abelian; in an abelian G every subgroup is normal, but an abelian subgroup of a non-abelian group need not be. Option C is Lagrange's Theorem, which constrains order and says nothing about normality. Option D describes H being abelian again, in different words.
The weaker choice: Option B is the most common error, and it comes from collapsing two different statements: 'in an abelian group all subgroups are normal' is true, 'an abelian subgroup is normal' is false. The course tests exactly this kind of precision — ILO2 asks you to identify examples and non-examples of normal subgroups, and the non-examples are where the marks are. watch this!
One exam decides 55% of your grade. Point-based marking. This whole page is built around that.
Overview
What MH2220 is, and where it sits
MH2220 is the course where mathematics becomes fully abstract. NTU states the aim plainly: to introduce group theory that is essential for more advanced algebra courses and applications, adding that the axiomatic concepts serve as a language to study concrete examples in a broader sense and help in developing logical thinking.
That phrase — axioms as a language — is the key to the course. A group is not a kind of number; it is any set with an operation satisfying four rules. Symmetries of a triangle, invertible matrices, permutations of a deck and the integers under addition are all the same object once you see them that way, and the course's central skill is recognising it.
The published thirteen-week schedule is a single continuous argument. Weeks 1 and 2 set the axioms and examples; week 3 introduces homomorphisms; weeks 4 to 6 build subgroups, cosets and Lagrange's Theorem; weeks 7 to 9 reach normal subgroups, quotient groups and the Isomorphism Theorems — three consecutive weeks, the course's summit; weeks 10 to 13 cover products, the classification of finite abelian groups, and group actions ending with the orbit-stabiliser theorem.
Always treat your own course outline and the exam timetable as authoritative.
Difficulty & time commitment
Is MH2220 hard, and how much time does it take?
MH2220 is manageable if you keep a weekly rhythm and treat the back half as the main event. The pattern is consistent: it starts gently and steepens, and the heaviest assessment is the part that separates grades.
The difficulty curve and the assessment weighting point the same way: the back half is harder and worth more. Front-loading effort there is the highest-return decision in the course.
Is this course for you
Who tends to do well, and who tends to struggle
You will likely do well if
- Your MH1300 proof skills are solid; this course is written in proofs from the first week.
- You work with the standard families — cyclic, dihedral, symmetric, matrix — until each is a reflex, since every abstract result is tested on them.
- You attend for the Wooclap sessions, though the best-five-of-ten rule gives real slack.
- You treat weeks 7 to 9 as the summit: normal subgroups, quotients and the Isomorphism Theorems are three consecutive weeks for a reason.
You may struggle if
- You try to compute your way through; there is almost nothing to compute and everything to justify.
- You learn definitions without non-examples — ILO2 asks specifically for examples and non-examples.
- You conflate 'abelian subgroup' with 'normal subgroup', which is the classic error.
- You miss a midterm over a timetable clash, which NTU explicitly does not accept.
- For every definition, hold one example and one non-example. ILO2 and ILO11 both ask for non-examples, and they are where precision is tested.
- Learn Lagrange, the Isomorphism Theorems and orbit-stabiliser as a connected chain rather than three separate results.
- Practise writing proofs in full prose; the tests and final are short-answer and marked on the argument.
- Work concrete groups by hand — build Cayley tables, compute cosets in S3 — before trusting the abstract statement.
Syllabus
The 12 topics, topic by topic
The exam-weight marker on each topic shows where the marks concentrate. The amber topics carry the highest exam weight.
T1 · Group axioms and examples
Week 1; ILO1, ILO5The four axioms, and why so many different objects satisfy them.
T2 · Properties of groups, Cayley tables and generators
Week 2; ILO1-4, ILO7Manipulating elements through generators and relations, and reading a Cayley table.
T3 · The standard families of groups
ContentCyclic, dihedral, symmetric, alternating, matrix and abelian groups — the examples everything is tested on.
T4 · Homomorphisms and isomorphisms
Week 3; ILO1, ILO4, ILO8Structure-preserving maps, and what it means for two groups to be the same.
T5 · Subgroups: centralizers, normalizers, stabilizers, kernels
Week 4; ILO1-3, ILO9The named subgroups that recur throughout group theory.
T6 · Cosets
Week 5; ILO1, ILO9Partitioning a group by a subgroup, the construction Lagrange's Theorem rests on.
T7 · Lagrange's Theorem
Week 6; ILO1, ILO9The order of a subgroup divides the order of the group, and what follows from it.
T8 · Normal subgroups and quotient groups
Week 7; ILO1-2, ILO9-10When a subgroup permits a quotient, and how to construct G/N.
T9 · The Isomorphism Theorems
Weeks 8-9; ILO1, ILO6, ILO8-9Two full weeks on the results that connect homomorphisms, kernels and quotients.
T10 · Direct and semidirect products
Week 10; ILO1-3, ILO6Building larger groups from smaller ones.
T11 · Classification of finite abelian groups
Week 11; ILO3, ILO6A complete classification — rare in mathematics, and worth understanding as such.
T12 · Group actions and the orbit-stabiliser theorem
Weeks 12-13; ILO3, ILO6, ILO11Groups acting on sets, ending with orbit-stabiliser and the Cauchy-Frobenius lemma.
How it's assessed
Assessment structure
| Component | Weight | Format & timing |
|---|---|---|
| Final examination (Short Answer Questions) | 55% | Individual in-person final examination of short answer questions, assessed against all eleven ILOs. Examination period. Point-based marking. |
| Short Answer Questions 1 | 20% | Individual in-person test covering ILOs 1, 2, 3, 4, 6 and 7. Mid-semester. Attendance mandatory; timetable clashes not accepted. |
| Short Answer Questions 2 | 20% | Individual in-person test covering ILOs 5, 8, 9, 10 and 11. Later in semester. Attendance mandatory; timetable clashes not accepted. |
| Class Participation | 5% | In-class Wooclap activities during lectures or tutorials. The best five of ten sessions count. Students discuss in groups but submit individual answers. During lectures and tutorials. Best 5 of 10 sessions. |
- The four components sum to 100. NTU states that attendance at midterm tests is mandatory and that timetable clashes are not accepted as a valid excuse — you are expected to plan your schedule accordingly. For medical emergencies you must provide a valid and verifiable medical certificate within the specified timeframe, and failure to comply may result in a grade penalty or a zero for the missed assessment.
- Ninety-five percent of the grade is timed and in person: a 55% final plus two 20% tests. The 5% participation component has a built-in buffer — the best five of ten Wooclap sessions count, so half can be missed — but that buffer does not extend to the tests, where NTU explicitly rules out timetable clashes as an excuse.
This is an exam-cram course. With the exams at 95% of the grade and the final examination (short answer questions) alone at 55%, your result is overwhelmingly decided by how well you perform under time pressure. Point-based marking.
Final exam timing: In-person short-answer final examination during the examination period. Confirm the exact date and venue on your exam timetable.
How to actually pass it
A weekly rhythm, two checklists, and the traps to avoid
The course rewards consistency over cramming, and practice over re-reading. Here is the loop that works, then what to have nailed before each exam.
The weekly loop
Before the mid-semester checklist
- State the group axioms and identify examples and non-examples of groups and subgroups.
- Manipulate elements using generators and relations, and interpret Cayley tables.
- Present permutations in different forms and compute orders and signatures.
- Prove and manipulate group homomorphisms and isomorphisms.
Before the final heaviest topics
- Compute cosets and apply Lagrange's Theorem.
- Identify normal subgroups and construct the quotient group G/N.
- Apply the Isomorphism Theorems and work with direct and semidirect products.
- Classify finite abelian groups, and apply group actions including the orbit-stabiliser theorem.
The mistakes that cost marks
Abelian subgroup assumed normal. In an abelian group every subgroup is normal; an abelian subgroup of a non-abelian group need not be. The two statements are constantly confused.
Lagrange read as an equivalence. Lagrange's Theorem says a subgroup's order divides the group's order. The converse — that a divisor gives a subgroup — is not generally true.
Definitions without non-examples. ILO2 and ILO11 both ask for non-examples. Knowing only what qualifies leaves half the assessment unprepared.
Timetable clash treated as an excuse. NTU states explicitly that clashes are not accepted for midterm absence. Plan the schedule before the semester starts.
Teaching team
Who teaches MH2220
The bios below are factual. We do not rate lecturers; any star ratings are submitted by students who have taken MH2220.
Teaching team as listed in public course information. AskSia does not rate lecturers; star ratings are submitted by students who have taken MH2220.
Formula & concept sheet
The vocabulary and formulas you must own
- Group
- A set with an associative operation, an identity, and inverses for every element.
- Cyclic group
- A group generated by a single element.
- Dihedral group
- The symmetry group of a regular polygon, including rotations and reflections.
- Symmetric and alternating group
- All permutations of a set, and the even permutations among them.
- Homomorphism
- A map between groups preserving the operation.
- Isomorphism
- A bijective homomorphism; two isomorphic groups are the same group in different notation.
- Kernel
- The elements a homomorphism sends to the identity; always a normal subgroup.
- Coset
- A translate of a subgroup by a group element; cosets partition the group.
- Lagrange's Theorem
- The order of a subgroup divides the order of a finite group.
- Normal subgroup
- A subgroup invariant under conjugation; exactly the condition permitting a quotient group.
- Quotient group
- The group formed by the cosets of a normal subgroup.
- Orbit-stabiliser theorem
- Relating the size of an orbit under a group action to the index of the stabiliser.
Common acronyms: AU · ILO · PLO.
Where it fits
Prerequisites, related courses & why it matters
Prerequisites published by NTU: (MH1200 or CY1602) and MH1300 — both linear algebra and Foundations of Mathematics are required. Mutually exclusive with MH2200. The course carries 3 Academic Units and 38 contact hours, offered in Year 2 Semester 2. The textbook is Dummit and Foote, Abstract Algebra, third edition.
Your MH2220 study toolkit
Study the course with Sia, not just read about it
Each tool already knows MH2220: your syllabus, your texts, and where the marks are. Grouped by how you study, from first contact to exam week.
FAQ
Frequently asked questions
Is MH2220 hard?
It rates hard. Abstract algebra is the sharpest transition in an undergraduate mathematics degree, and 95% of the grade is in timed in-person assessment with a 55% final.
What is the assessment breakdown?
Final examination 55%, Short Answer Questions 1 20%, Short Answer Questions 2 20%, class participation 5%. All are point-based marking.
How does the participation component work?
In-class Wooclap activities during lectures or tutorials, with the best five of ten sessions counting. You may discuss in groups but must submit individual answers. Missing half the sessions costs nothing.
What if I have a timetable clash with a midterm?
NTU addresses this directly: timetable clashes are not accepted as a valid excuse for absence, and you are expected to plan your schedule accordingly. Only a valid and verifiable medical certificate, submitted within the specified timeframe, is accepted.
What do I need before taking it?
Both (MH1200 or CY1602) and MH1300. The Foundations prerequisite matters as much as the linear algebra — this course is written in proofs from week one.
Who wrote the course?
The course author published in the OBTL document is Lim Kay Jin, Division of Mathematical Sciences.
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