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MTH312Sciences3 Unitsintermediate

Abstract Algebra Ii

This course, Groups and Rings, builds upon concepts introduced in MTH 211, elaborating on subgroups with specific characteristics termed normal subgroups. It explores algebraically indistinguishable systems through isomorphism, a concept first used by Camille Jordan. Isomorphisms are presented as special cases of homomorphisms, functions preserving algebraic structure. The course extends these concepts to Ring Theory, defining rings, sub-rings, and various types thereof. Ring homomorphism and isomorphisms are also covered, mirroring the approach used in group theory.

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200h
Study Time
13
Weeks
15h
Per Week
intermediate
Math Level
Course Keywords
GroupsRingsHomomorphismIsomorphismIdeals

Course Overview

Everything you need to know about this course

Course Difficulty

Intermediate Level
Builds on foundational knowledge
65%
intermediate
📊
Math Level
Moderate Math
📖
Learning Type
Theoretical Focus

Course Topics

Key areas covered in this course

1

Normal Subgroups

2

Group Homomorphisms

3

Permutation Groups

4

Finite Groups

5

Rings

6

Ideals

7

Ring Homomorphisms

8

Quotient Rings

9

Isomorphism Theorems

10

Sylow Theorems

Total Topics10 topics

Requirements

Knowledge and skills recommended for success

MTH 211

💡 Don't have all requirements? Don't worry! Many students successfully complete this course with basic preparation and dedication.

Assessment Methods

How your progress will be evaluated (3 methods)

Assignments

Comprehensive evaluation of course material understanding

Written Assessment

Tutor Marked Assignments

Comprehensive evaluation of course material understanding

Written Assessment

Final Examination

Comprehensive evaluation of course material understanding

Written Assessment

Career Opportunities

Explore the career paths this course opens up for you

Pure Mathematics Researcher

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Data Scientist

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Cryptography

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Actuarial Science

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Statistician

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Industry Applications

Real-world sectors where you can apply your knowledge

TelecommunicationsData SecurityFinanceAcademiaResearch

Study Schedule Beta

A structured 13-week journey through the course content

Week
1

Module 1:

3h

Unit 1: Normal Subgroups

3 study hours
  • •Understand the definition of normal subgroups and their properties.
  • •Verify if a subgroup is normal.
  • •Learn to obtain a quotient group corresponding to a given normal subgroup.
Week
2

Module 1:

4h

Unit 2: Group Homomorphisms

4 study hours
  • •Study group homomorphisms and their properties.
  • •Learn to obtain the kernel and image of a homomorphism.
  • •Understand the concept of isomorphism and how to check if a function is an isomorphism.
Week
3

Module 1:

3h

Unit 3: Permutation Group

3 study hours
  • •Understand the definition of permutation groups.
  • •Learn to express any permutation as a product of disjoint cycles.
  • •Find out whether an element is odd or even.
Week
4

Module 1:

3h

Unit 4: Finite groups

3 study hours
  • •Study finite groups and their properties.
  • •Learn to determine the possible subgroups and structures of finite groups using Sylow's theorems.
  • •Classify groups of order p, p2 or pq where p and q are primes.
Week
5

Module 2:

4h

Unit 1: Rings

4 study hours
  • •Understand the definition of a ring and its properties.
  • •Learn about elementary properties of rings.
  • •Study different types of rings, such as commutative rings and rings with identity.
Week
6

Module 2:

4h

Unit 2: Subrings and Ideals

4 study hours
  • •Study subrings and ideals.
  • •Learn to check whether a subset of a ring is a subring or an ideal.
  • •Understand the properties of ideals and their role in forming quotient rings.
Week
7

Module 2:

4h

Unit 3: Ring Homomorphisms

4 study hours
  • •Understand ring homomorphisms and their properties.
  • •Learn to obtain the kernel and image of a ring homomorphism.
  • •Study the Isomorphism Theorems and their applications.
Week
8

Module 1:

4h

Module 1 Review

4 study hours
  • •Review Module 1: Groups
  • •Focus on key concepts: Normal Subgroups, Group Homomorphisms, Permutation Groups, Finite Groups
Week
9

Module 2:

4h

Module 2 Review

4 study hours
  • •Review Module 2: Rings
  • •Focus on key concepts: Rings, Subrings and Ideals, Ring Homomorphisms
Week
10

Module 1:

3h

Additional Exercises: Normal Subgroups and Homomorphisms

3 study hours
  • •Solve additional exercises on normal subgroups and quotient groups.
  • •Practice problems on group homomorphisms and isomorphisms.
Week
11

Module 2:

3h

Additional Exercises: Rings and Homomorphisms

3 study hours
  • •Work through examples on rings, subrings, and ideals.
  • •Practice problems on ring homomorphisms and isomorphisms.
Week
12

Modules 1 & 2

4h

TMA Preparation and Submission

4 study hours
  • •Complete all Tutor Marked Assignments (TMAs) for Modules 1 and 2.
  • •Ensure all assignments are submitted on time.
Week
13

Modules 1 & 2

4h

Final Revision

4 study hours
  • •Final revision of all course materials.
  • •Focus on areas of weakness identified during the semester.

This study schedule is in beta and may not be accurate. Please use it as a guide and consult the course outline for the most accurate information.

Course PDF Material

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Study Tips & Exam Preparation

Expert tips to help you succeed in this course

1

Create detailed concept maps linking normal subgroups, quotient groups, homomorphisms, and isomorphisms.

2

Practice proving whether given subgroups are normal and constructing corresponding quotient groups.

3

Focus on applying Sylow's Theorems to determine possible subgroup structures of finite groups.

4

Review definitions and properties of rings, subrings, ideals, and their interrelationships.

5

Practice solving problems involving ring homomorphisms, kernel/image calculations, and applying the Fundamental Theorem.

6

Work through numerous examples of quotient ring constructions and their properties.

7

Review all Tutor Marked Assignments (TMAs) and focus on areas where marks were lost.

8

Allocate specific time slots for focused study sessions each week, avoiding last-minute cramming.

9

Form a study group to discuss challenging concepts and practice problem-solving collaboratively.

10

Prioritize understanding over memorization; focus on the underlying principles and relationships between concepts.

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