Abstract Algebra Ii
- Sciences
- 300 level
- 3 credit units
- 151 pages
- 7 units
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.
About this course
- Difficulty
- Intermediate
- Study hours
- 200 hours
- Maths
- Intermediate
- Content
- Theoretical, problem solving
- Practical work
- No
- MTH 211
- Assignments
- Tutor Marked Assignments
- Final Examination
What you'll read
The real module and unit structure of MTH312, taken from the course material NOUN publishes.
One paragraph, so you can see how it reads
MTH312 · UNIT 1 NORMAL SUBGROUPS
Example 1 is a special case of the fact that every subgroup of a commutative group is a normal subgroup. We will prove this fact later (in Theorem 2).
What you should be able to do
- Define normal subgroups and quotient groups.
- Apply group homomorphism and isomorphism concepts.
- Solve problems related to permutation groups.
- Apply Sylow's theorems to analyze finite groups.
- Define rings, sub-rings, and ideals.
- Apply ring homomorphism and isomorphism concepts.
- Construct quotient rings and understand their properties.
What it prepares you for
- Pure Mathematics Researcher
- Data Scientist
- Cryptography
- Actuarial Science
- Statistician
- Telecommunications
- Data Security
- Finance
- Academia
- Research
Where it gets hard
The units students slow down on, and what makes each one heavy.
- Module 1:
Unit 1: Normal Subgroups
Requires understanding of subgroups and cosets from MTH 211, and the ability to verify whether a subgroup is normal or not.
- Module 1:
Unit 2: Group Homomorphisms
Requires understanding of abstract algebraic structures and the ability to verify whether a function between groups is a homomorphism or not.
A suggested way through it
13 weeks, about 47 hours in total. Yours will differ.
- Week 1Module 1:
Unit 1: Normal Subgroups · 3 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 2Module 1:
Unit 2: Group Homomorphisms · 4 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 3Module 1:
Unit 3: Permutation Group · 3 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 4Module 1:
Unit 4: Finite groups · 3 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 5Module 2:
Unit 1: Rings · 4 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 6Module 2:
Unit 2: Subrings and Ideals · 4 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 7Module 2:
Unit 3: Ring Homomorphisms · 4 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 8Module 1:
Module 1 Review · 4 hours
Review Module 1: Groups. Focus on key concepts: Normal Subgroups, Group Homomorphisms, Permutation Groups, Finite Groups.
- Week 9Module 2:
Module 2 Review · 4 hours
Review Module 2: Rings. Focus on key concepts: Rings, Subrings and Ideals, Ring Homomorphisms.
- Week 10Module 1:
Additional Exercises: Normal Subgroups and Homomorphisms · 3 hours
Solve additional exercises on normal subgroups and quotient groups.. Practice problems on group homomorphisms and isomorphisms..
- Week 11Module 2:
Additional Exercises: Rings and Homomorphisms · 3 hours
Work through examples on rings, subrings, and ideals.. Practice problems on ring homomorphisms and isomorphisms..
- Week 12Modules 1 & 2
TMA Preparation and Submission · 4 hours
Complete all Tutor Marked Assignments (TMAs) for Modules 1 and 2.. Ensure all assignments are submitted on time..
- Week 13Modules 1 & 2
Final Revision · 4 hours
Final revision of all course materials.. Focus on areas of weakness identified during the semester..
Preparing for the exam
- Create detailed concept maps linking normal subgroups, quotient groups, homomorphisms, and isomorphisms.
- Practice proving whether given subgroups are normal and constructing corresponding quotient groups.
- Focus on applying Sylow's Theorems to determine possible subgroup structures of finite groups.
- Review definitions and properties of rings, subrings, ideals, and their interrelationships.
- Practice solving problems involving ring homomorphisms, kernel/image calculations, and applying the Fundamental Theorem.
- Work through numerous examples of quotient ring constructions and their properties.
- Review all Tutor Marked Assignments (TMAs) and focus on areas where marks were lost.
- Allocate specific time slots for focused study sessions each week, avoiding last-minute cramming.
- Form a study group to discuss challenging concepts and practice problem-solving collaboratively.
- Prioritize understanding over memorization; focus on the underlying principles and relationships between concepts.
Questions students ask about this course
What is MTH312 about?
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.
How many units does MTH312 have?
MTH312, Abstract Algebra Ii, has 7 units across 2 modules, over 151 pages of course material. You can read it one unit at a time.
How many credit units is MTH312?
MTH312 carries 3 credit units, at 300 level in Sciences.
Is MTH312 hard?
MTH312 is rated intermediate level, with intermediate mathematical content. It is mostly theoretical and problem solving work.
How long does MTH312 take to study?
About 200 hours of study, spread across its 7 units.
How is MTH312 assessed?
MTH312 is assessed by Assignments, Tutor Marked Assignments and Final Examination.
What do I need before starting MTH312?
MTH 211
What can I do with MTH312?
Pure Mathematics Researcher, Data Scientist, Cryptography, Actuarial Science and Statistician.