SMA 3205 - Introduction to Algebra
An introductory undergraduate course exploring the foundational concepts of modern abstract algebra. The course equips students with structural tools to analyse binary operations, the integers, and group theory, providing a bridge to advanced mathematical concepts, coding theory, and cryptography.
Course learning outcomes
By the end of this course, you should be able to:
- Analyse fundamental algebraic structures, including binary operations, equivalence relations, and integers, to construct rigorous mathematical proofs
- Apply the core axioms of group theory to identify, classify, and perform computations within specific group families such as cyclic, permutation, and dihedral groups
- Evaluate the relationships between groups, subgroups, and cosets using key theorems (e.g., Lagrange’s Theorem) to solve complex structural problems.
- Synthesise the conceptual hierarchy of abstract algebra by extending group properties to Rings and Fields, and connecting these structures to practical applications.
Course information | SMA 3205 - Introduction to Algebra | 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: Relations and Binary Operations
By the end of this week, you should be able to: * Define binary relations, identify their properties, and provide structural examples. * Analyse equivalence relations, determine equivalence classes, and explain how they partition sets. * Formulate the definition of binary operations and construct concrete examples. * Verify the core properties of binary operations, including closure, commutativity, associativity, identity elements, and inverses.
- 1.1. Binary relations
- 1.2. Equivalence relations
- 1.3. Definition and concrete examples of binary operations.
- 1.4. Core properties of binary operations
- Week 2: The Integers and Factorisation
- 2.1. Introduction to Rings and Fields through the lens of the integers.
- 2.2. Divisibility, the Euclidean algorithm, and greatest common divisors.
- 2.3. Prime numbers, irreducibility, and prime factorisation.
- 2.4. The Fundamental Theorem of Arithmetic.
- Week 3: Congruences and Cryptographic Applications
- 3.1. Modular arithmetic, definitions of congruences, and their algebraic properties.
- 3.2. Constructing and computing with congruence classes.
- 3.3. Exploring algebraic applications in Coding Theory.
- 3.4. Utilising modular arithmetic and integers for basic Cryptography applications.
- Week 4: Introduction to Groups
- 4.1. Formal definition of a Group and fundamental axioms.
- 4.2. Deriving basic properties of group elements (uniqueness of identity and inverses, cancellation laws).
- 4.3. Exploring concrete examples of discrete and continuous groups.
- 4.4. The concept of the order of a group and the order of an individual element.
- Week 5: Subgroups and Group Maps
- 5.1. Defining Subgroups and establishing subgroup tests (one-step and two-step criteria).
- 5.2. Identifying and analysing subgroups within known group structures.
- 5.3. Introduction to Group Maps: Defining and exploring Homomorphisms.
- 5.4. Concept of Isomorphisms: Showing when two groups are structurally identical.
- Week 6: Cyclic Groups
- 6.1. Formal definition and properties of Cyclic Groups.
- 6.2. Identifying generators and analysing the structure of cyclic subgroups.
- 6.3. The Fundamental Theorem of Finite Cyclic Groups.
- 6.4. Classifying cyclic groups up to isomorphism and determining subgroups of cyclic groups.
- Week 7: Permutation Groups
- 7.1. Introduction to permutations and the formal definition of the Symmetric Group
- 7.2. Utilising cycle notation, disjoint cycles, and transposition computations.
- 7.3. Analysing even and odd permutations, and defining the Alternating Group
- 7.4. Structural properties and applications of permutation groups in abstract algebra.
- Week 8: Dihedral Groups and Symmetries
- 8.1. Geometric interpretations of algebra: symmetries of regular polygons.
- 8.2. Defining Dihedral groups (Dn) via generators (rotations and reflections) and relations.
- 8.3. Executing group operations and analysing the non-abelian nature of Dihedral groups.
- 8.4. Comparing the subgroups and structural behaviours of Sn and Dn.
- Week 9: Cosets and Lagrange’s Theorem
- 9.1. Defining left and right cosets of a subgroup.
- 9.2. Establishing the properties of cosets, including disjointness and equivalent sizes.
- 9.3. Stating and proving Lagrange’s Theorem for finite groups.
- 9.4. Direct corollaries of Lagrange’s Theorem (e.g., classifying groups of prime order).
- Week 10: Applications of Lagrange's Theorem
- 10.1. Calculating the index of a subgroup.
- 10.2. Using Lagrange's Theorem to solve structural problems within finite groups.
- 10.3. Fermat’s Little Theorem and Euler’s Theorem as algebraic consequences.
- 10.4. Connecting group homomorphisms with subgroups and cosets (intro to kernels).
- Week 11: Introduction to Rings
- 11.1. Extending abelian groups with a second binary operation: Axioms of a Ring.
- 11.2. Basic properties of rings and subrings.
- 11.3. Examples of rings: The integers Z, modulo rings Zn, matrix rings, and polynomial rings.
- 11.4. Commutative rings, rings with unity, and properties of units and zero divisors.
- Week 12: Integral Domains and Fields
- 12.1. Defining Integral Domains and their cancellation properties.
- 12.2. Introduction to Fields: Definition and classic examples (Q, R, C, Zp).
- 12.3. Proving that every finite integral domain is a field.
- 12.4. Reviewing the hierarchy of algebraic structures (Groups ⊂ Rings ⊂ Fields).
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