Degree Programme Course
SMA 3405 - Algebra II
The credits you earn are recognised academic credits
Advanced undergraduate introduction to Abstract Algebra. This course equips students with structural tools useful in group theory, Galois theory, and ring theory, bridging abstract algebraic foundations with concrete computational and structural problem-solving techniques.
Course learning outcomes
By the end of this course, students will be able to:
- Construct advanced algebraic structures, including direct products of groups.
- Apply group actions, the Orbit-Stabilizer theorem, and Sylow theorems to solve structural classification problems and combinatorial enumeration tasks.
- Examine normal series, factor groups, and composition series in solvability and structural properties of finite groups.
- Demonstrate mastery of ring theory and field extensions by executing polynomial division, analysing ideal domains, and constructing finite fields.
Prerequisites
SMA 3303 - Algebra I
First published: 16/08/26Updated: 29/08/26
Course information | SMA 3405 - Algebra II | SOMAS
Below is the course content. You can click on any section here and it will take you through to this section of the course. If you are signed in and enrolled on this course we can track your progress.
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- Week 1: Review of Core Algebraic Structures and Group Examples
- 1.1. Review of basic group concepts, commutative groups Zn, and the group of units under multiplication modulo n
- 1.2. Symmetric groups, signature homomorphisms, and alternating groups as normal subgroups
- 1.3. Dihedral groups as symmetry groups of regular polygons, defined via generators and relations
- 1.4. Matrix groups including the general linear group and special linear group over commutative rings
- Week 2: Direct Products and Abelian Groups
- 2.1. Constructing new groups from known groups using direct products and examining their structural properties
- 2.2. Analysing element orders, generators, and cyclic structures within direct product configurations
- 2.3. Investigating decomposition patterns and structural classifications of finite abelian groups
- 2.4. Summary of Week 2 concepts and guided introductory exercises
- Week 3: Structure of Finite Abelian Groups
- 3.1. Introduction to the Fundamental Theorem of Finitely Generated Abelian Groups
- 3.2. Analysing invariant factors, elementary divisors, and primary decomposition methods
- 3.3. Applying theoretical classifications to compute isomorphism classes of finite abelian groups
- 3.4. Summary of Week 3 and collaborative structural problem-solving session
- Week 4: Introduction to Group Actions
- Week 1: Introduction to Group Actions
- 1.1. Formal definition and foundational concepts of a group acting on a set
- 1.2. Analysing orbits, stabilizers, and establishing the Orbit-Stabilizer Theorem
- 1.3. Connecting abstract group actions to geometric visualizations of spatial symmetries and polygon transformations
- 1.4. Practical applications to spatial symmetries and configuration counting
- Week 2: Conjugacy Classes and Class Equation
- 2.1. Investigating conjugacy classes and their fundamental algebraic properties within a group
- 2.2. Deriving the Class Equation and examining its structural consequences for finite groups
- 2.3. Exploring the centre of a group and structural constraints governing $p$-groups
- 2.4. Summary of Week 5 and targeted computational exercises
- Week 3: Applications of Group Actions: Burnside’s Theorem & Sylow Theorems
- 3.1. Utilising Burnside’s Theorem (Cauchy-Frobenius lemma) to solve combinatorial enumeration and colouring problems
- 3.2. Introduction to the First, Second, and Third Sylow Theorems for finite groups
- 3.3. Applying Sylow theorems to determine the non-existence of simple groups of specific orders
- 3.4. Combinatorial enumeration workshops and advanced problem-solving
- 1. Factor Groups and Normal Series
- 1.1. Review of fundamental isomorphism theorems and the properties of normal subgroups and quotient groups
- 1.2. Defining and constructing subnormal and normal series for groups
- 1.3. Analysing factor groups and inherited quotient structure properties
- 1.4. Summary of Week 7 and structural proof exercises
- 2. Composition Series and the Jordan-Hölder Theorem
- 2.1. Defining composition series, refinement of series, and simple groups
- 2.2. Proving and applying the Jordan-Hölder Theorem to establish the uniqueness of composition factors
- 2.3. Comparative analysis of composition factors across different non-abelian group families
- 2.4. Applications to the classification framework of finite groups
- 3. Solvable and Simple Groups
- 3.1. Defining and characterising solvable groups through derived series and abelian factors
- 3.2. Investigating commutator subgroups and their role in measuring non-commutativity
- 3.3. Proving the simplicity of alternating groups.
- 3.4. Summary of Week 9 and advanced proof techniques in structural algebra
- 1. Rings and Polynomial Rings
- 1.1. Review of ring axioms, integral domains, commutative rings with unity, and fields
- 1.2. Investigating polynomial rings R[x] over commutative base rings
- 1.3. Executing the division algorithm and analysing ideal structures within polynomial rings
- 1.4. Summary of Week 10 and computational examples of ring homomorphisms
- 2. Euclidean Domains and Principal Ideal Domains
- 2.1. Defining Euclidean norms and identifying Euclidean domains
- 2.2. Exploring Principal Ideal Domains (PIDs) and Unique Factorisation Domains (UFDs)
- 2.3. Analysing irreducible elements, prime elements, and ideal containment chains
- 2.4. Practical applications to polynomial factorisation and divisibility testing
- 3. Field Extensions and Finite Fields
- 3.1. Introduction to field extensions, splitting fields, and algebraic elements
- 3.2. Calculating the degree of an extension and determining minimal polynomials
- 3.3. Constructing and analysing the uniqueness of finite fields
- 3.4. Comprehensive course summary and final review
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