BSc Computer Science and Mathematics with Industrial Experience

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Algebra is one of the fundamental areas of mathematics. Through algebraic structures common to many different areas, it allows to formalise and link disparate mathematical ideas as well as to study them in great depth. Three of the most important and ubiquitous structures are groups, rings and modules. We build to two of the fundamental results in the algebra: Sylow’s theorems for finite groups; and the structure theorem for modules over principal ideal domains, as well giving a glimpse into the world of finite simple groups.

The unit will build on previous units with algebraic content, namely Linear Algebra, Groups and Geometry, and Rings and Fields, to explore group theory, ring theory and linear algebra in greater depth. It will expose students to fundamental and interesting results in the subject, whilst laying foundation for further study. 

• Define standard subgroup constructions such as normalizer, centralizer and subgroup generated by a set, and prove elementary properties. Calculate these objects in small examples.  
• State, prove and apply Sylow’s theorems.
• Define and apply basic properties of commutative rings.
• Define and provide examples of basic notions relating to modules and homomorphisms
• Compute Smith normal form and apply it to determine the isomorphism type modules over PID’s and to compute the rational canonical form
• Prove simple unseen statements involving the various algebraic structures discussed in this unit
• Recognise examples of simple groups and perform elementary calculations in them.  
• Define and compute composition series in specific examples and state and apply the Jordan-Hölder theorem.

• Properties of commutative rings, in particular integral domains, PID’s and Euclidean domains with Z and C[x] as main examples. Show that Euclidean domains are UFD’s [2 weeks]
• Definition of modules, submodules, generators, cyclic modules, homomorphisms, isomorphisms, direct sums. [2 weeks]
• Examples of modules: modules over CG, modules over C[X] as vector spaces with a distinguished endomorphism. Definition of torsion modules and free modules with examples [1 week]
• Smith normal form. Statement and proof of the structure theorem for finitely generated modules over PID’s. Deduce the classification of finite abelian groups. [4 weeks]
• Relationship between of matrices and finitely generated C[x]-modules, and deduce the rational canonical form [2 weeks]
• Simple groups. Composition series and the Jordan-Hölder theorem. [2 weeks]
• Simplicity of Alternating groups and PSLn(F) when n ≥2, |F| ≠ 2,3. Introduction to the classification of finite simple groups. [3 weeks] 

The content will be delivered through lectures.  

There will be 3 hours of contact time per week, consisting of two lectures and a weekly 1-hour tutorial.  

There will be two formative exercises, one in each semester based on foundational material from each part of the unit.

Script viewing and general feedback available after marks are released

• John B Fraleigh, A First Course in Abstract Algebra, (5th edition), 1967, Addison-Wesley.
• Brian Hartley, Trevor O. Hawkes, Rings, Modules and Linear Algebra, 1974, Chapman and Hall Limited.