MTH6106 - Group Theory - 2023/24
Topic outline
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Year: 3 | Semester: A | Level: 6| Credits: 15Class schedule:
This module is taught in a "3+1" format, with three lectures per week and one tutorial, as follows:
- Lectures 1&2: Room GC202 (Graduate Centre), Tuesdays 14:00-16:00
- Lecture 3 & Tutorial: Room GC604 (Graduate Centre), Thursdays 14:00-16:00.
Unfortunately these rooms are not equipped with Q-Review.
Final Marks: 80% exam, 20% coursework
Module Organiser: Ian Morris
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This handwritten assessment is available for a period of 3 hours and 30 minutes in total, within which you must submit your solutions. This time frame includes 30 minutes for scanning and submitting your solution. You may log out and in again during that time, but the countdown timer will not stop. If your attempt is still in progress at the end of the set time slot, any file you have uploaded will be automatically submitted.
The assessment is intended to be completed within 3 hours.
IMPORTANT – If you run out of time during this assessment, we will not accept IT issues as an extenuating circumstance. This is because we have given you ample time to complete and submit your work. If your attempt is still in progress at the end of the set time slot, any file you have uploaded will be automatically submitted.
In completing this assessment:
• You may use books and notes.
• You may use calculators and computers, but you must show your working for any calculations you do.
• You may use the Internet as a resource, but not to ask for the solution to an exam question or to copy any solution you find.
• You must not seek or obtain help from anyone else.
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This week's lectures will cover the following fundamental definitions of group theory: groups; subgroups; the order of a group; the order of an element. We will also introduce Cayley tables, introduce the idea of a subgroup generated by a set of elements, and consider some simple examples of groups.
This week's classes will correspond roughly to Sections 1.0 to 1.3 of the lecture notes.
At the end of this week you will be able to use the Week 1 quiz. This quiz does not carry credit, and may be taken any number of times. -
This week's focus will be on familiarising ourselves with examples of groups, including: cyclic groups; the Quaternion group; groups defined by multiplication modulo an integer; dihedral groups; and some examples of groups of matrices. We may also begin to investigate the symmetric group.
This week's material will correspond approximately to sections 1.4 to 2.5 of the printed notes.
The first coursework will be issued on Thursday at 4pm this week, with a deadline of 2pm on the following Friday.-
The coursework will be available for download from 4pm on Thursday Week 2.
This coursework counts for 4% of your mark for this module. You should answer all questions, and each question will be marked out of 4. You should give full explanation of your answers. Please submit your solutions on QMPlus by 2pm on Friday 13th October (Week 3). Your submission must be entirely your own work.
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This week's objectives will be to study the symmetric and alternating groups, which are certain groups of permutations of finite sets. We will consider two-line notation and disjoint cycle notation for permutations, and familiarise ourselves with converting between these two notations. We will also show that the alternating group is a subgroup of the symmetric group.
Towards the end of the week we will define the cosets of a subgroup of a group and begin to investigate the properties of cosets.
This week's classes correspond roughly to sections 2.6 and 3.1 of the lecture notes.
At the end of this week you will be able to use the Week 3 quiz. -
This module is about the theory of groups, continuing from the basics taught in Introduction to Algebra. Groups describe `symmetry’ and for that matter they arise all over mathematics. We will delve further into the study of groups (from a rather abstract viewpoint), and in so doing meet many more examples of groups including ones arising in geometry and from other algebraic objects such as number systems and matrices. In many scenarios groups operate on certain sets (think of the group of symmetry operations of some geometric shape for instance). The theory of group actions deals with this kind of situation. Another highlight of the module are Sylow's Theorems which provide precise information on the number of subgroups of certain orders that a group must have.
Throughout this module there is a strong emphasis on abstract thinking and proof.
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This week we investigate the notion of coset of a subgroup and begin to prove some facts about cosets, including Lagrange's Theorem, the first powerful result proved in this course. We then go on to investigate the notion of conjugacy in a group, and we define the centre subgroup of an arbitrary group. We conclude the week's work by studying conjugacy of elements in the specific context of the symmetric group.
This week's classes should correspond approximately to sections 3.1 to 3.3 of the notes.
The second coursework will be issued this week, with a deadline on the following Friday.-
This coursework will be available for download from 4pm on Thursday in Week 4.
This coursework counts for 4% of your mark for this module. You should answer all questions, and each question will be marked out of 4. You should give full explanation of your answers. Please submit your solutions on QMPlus by 2pm on Friday 27th October (Week 5). Your submission must be entirely your own work.
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This week we consider conjugacy of subgroups (as opposed to conjugacy of elements, which we looked at last week) and go on to define: normal subgroups; the quotient of a group by a normal subgroup; products of groups. We'll illustrate these last three topics with examples.
This material corresponds roughly to sections 3.4 to 3.6 of the notes.At the end of this week you will be able to use the Week 5 quiz.
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This week we will wrap up Chapter 3 of the notes by considering the commutator subgroup of a group. The notion of commutator subgroup depends crucially on the idea of a subgroup generated by a set of elements, which we introduced in the first week of the course. We then go on to introduce homomorphisms, which are a mechanism allowing us to compare and relate the structure of one group with the structure of another group.
This week's activity will correspond roughly to sections 3.7 to 4.2 of the course notes.
The third coursework will be issued this week, with a deadline on the following Friday.-
This coursework will be available for download from 4pm on Thursday in Week 6.
Update: as of Tuesday 7th I have uploaded a corrected version which amends a typo in Question 4 (thanks to the person who spotted this!).
This coursework counts for 4% of your mark for this module. You should answer all questions, and each question will be marked out of 4. You should give full explanation of your answers. Please submit your solutions on QMPlus by 2pm on Friday 10th November (Week 7). Your submission must be entirely your own work.
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As is standard practice at QMUL, there will be no scheduled teaching activities in the seventh week of the semester.
I will be on annual leave on the Wednesday, Thursday and Friday of this week. -
This week we will prove the First, Second and Third Isomorphism Theorems, which are powerful general results connecting homomorphisms, normal subgroups, and quotient groups. These theorems will allow us to understand the notion of quotient group, which was introduced in Week 5 via explicit examples, in a systematic way. This week's material is the most abstract material occurring in the course, but its results -- especially the First Isomorphism Theorem -- will be essential in certain later parts of the course. We also briefly study the notion of a group automorphism and introduce inner and outer automorphism groups.
This week's activity corresponds roughly to sections 4.2 and 4.3 of the notes.
At the end of this week you will be able to use the Week 8 quiz. -
This week we will introduce the notion of an action of a group, which is of fundamental importance when applying group theory to problems in other areas of mathematics such as geometry and combinatorics. We will introduce the notions of orbit and stabiliser and prove the Orbit-Stabiliser Theorem, which is one of the most useful results in the course in terms of applications. We will consider some applications of the orbit-stabiliser theorem to counting problems in combinatorics and linear algebra, and also apply it to study the symmetry groups of geometric objects. This activity may continue into the early part of Week 10.
This week we expect to cover most or all of section 5 of the notes.
The fourth coursework will be issued this week, with a deadline on the following Friday.-
This coursework will be available for download from 4pm on Thursday in Week 9.
This coursework counts for 4% of your mark for this module. You should answer all questions, and each question will be marked out of 4. You should give full explanation of your answers. Please submit your solutions on QMPlus by 2pm on Friday 1st December (Week 10). Your submission must be entirely your own work.
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This week we will finish our investigation of applications of the Orbit-Stabiliser Theorem and move on to study simple groups, which are important in the structure theory of finite groups. We will finally define and begin to investigate composition series. Composition series are a means of expressing an arbitrary finite group in terms of simple groups, in a way which is somewhat analogous to the fact that an arbitrary integer can be expressed as a product of primes. (Composition series will later be important in the Year 7 module MTH745 Further Topics in Algebra.)
This week we expect to proceed from the end of section 5 of the notes to the beginning of section 6.3.
At the end of this week you will be able to use the Week 10 quiz. -
This week we conclude the investigation of composition series and begin to investigate p-groups, which are groups whose order is a power of a prime number. Our first main task in this area will be to classify all groups whose order is either a prime number or a square of a prime number.
This week we will cover roughly from section 6.3 to section 7.2 in the notes.
The final coursework will be issued this week, with a deadline on the following Friday. -
- There will be five assessed courseworks for this module.
- Each coursework counts for 4% of your module mark.
- You will have a week to complete each coursework.
- Your work should be entirely your own, and must be submitted on QMplus.
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Format of the final exam
The final exam for MTH6106 in 2023/24 will be conducted online. You will have a four-hour window in which to complete the examination.
Past papers
Exam papers for the last ten years are available below, to help you prepare, but bear in mind the differences you'll see with an online exam. Also, a few other comments:
- Prior to 2020, the module was called MTH6104 Algebraic Structures II. The content was exactly the same: only the name and code changed.
- The syllabus has been shortened slightly over the years, so a few of the questions from the past papers are not relevant; so please ignore question 6(b–e) on the 2014 and 2017 papers.
- In 2017 and earlier the exam rubric was different: you could choose which questions to answer. The rubric for this module is now "answer all questions", as for most other undergraduate exams.
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- Introduction and basics
Definition of group, subgroup, order, generators, and basic properties. - Examples
Cyclic gropus, the quaternion group of order 8, symmetric groups, alternating groups, symmetry group of geometric objects, matrix groups, group of units modulo n. - Cosets and Conjugacy
Cosets, Lagrange's Theorem, conjugacy, normal subgroups, quotient groups, products of subgroups, the commutator subgroup. - Homomorphisms
Definitions, image and kernel, the Isomorphism Theorems, the Correspondence Theorem, automorphisms, inner and outer automorphism groups. - Actions
Definitions, orbits, stabilisers, the Orbit–Stabiliser Theorem, centralisers and normalisers, the Orbit-Counting Lemma and applications. - Simple groups and composition series
Definitions, simple abelian groups, simplicity of alternating groups, composition series, statement of Jordan-Hölder. - p-groups
Sylow p-subgroups, the Sylow theorems, finite p-groups.
- Introduction and basics
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At the end of this module, students should be able to:
- Define groups, homomorphisms, (normal) subgroups, quotient groups and related objects.
- Define and compute with examples, including the quaternion group, dihedral groups, the symmetric group and matrix groups.
- State and prove basic properties of groups, including Lagrange's Theorem and the Isomorphism Theorems.
- State and (with guidance) prove more advanced results concerning groups.
- Define group actions, work with examples, and prove basic results.
- Create examples to illustrate the underlying theory, and work with these examples.
- Recount and explain concepts of group theory.
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These quizzes don't count towards your final mark - they're just to help you learn. They all have multiple-choice or numerical answers which are marked automatically, and you can attempt them as many times as you want. So please keep trying until you get all the questions right!
Some of the questions are deliberately quite tough, because you're allowed several attempts, so don't get disheartened if you find it difficult - the best way to learn is to struggle with difficult tasks.
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This quiz will test you on the concepts taught during week 1. It will become available at the end of the week.
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This quiz will test you on the concepts taught up to the end of week 3. It will become available at the end of the week.
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This quiz will test you on the concepts taught up to the end of week 5. It will become available at the end of the week.
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This quiz will test you on the concepts taught up to the end of week 8. It will become available at the end of the week.
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This quiz will test you on the concepts taught up to the end of week 10. It will become available at the end of the week.
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Lecture notes for this module are available here.
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Recall that by Lagrange's theorem, if a group of order n has a subgroup of order m, then m must divide n. This week we are motivated by the reverse question: if a group has order n, and if m is an integer which divides n, does the group necessarily have a subgroup of order m? In this week's classes we will study Sylow's Theorems, which give a positive answer to this question in certain cases. We will also apply Sylow's theorems to the classification of finite simple groups.
On the Tuesday of this week we will cover roughly sections 7.3 to 7.4 of the notes, and on the Thursday we will go through the 2021 exam paper (provided below). -
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Unfortunately this module is taught in rooms which are not equipped with Q-Review.
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