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This subject introduces the theory of groups, which is at the core of modern algebra, and which has applications in many parts of mathematics, chemistry, computer science and theoretical physics. It also develops the theory of linear algebra, building on material in earlier subjects and providing both a basis for later mathematics studies and an introduction to topics that have important applications in science and technology.
Topics include: modular arithmetic and RSA cryptography; abstract groups, homomorphisms, normal subgroups, quotient groups, group actions, symmetry groups, permutation groups and matrix groups; theory of general vector spaces, inner products, linear transformations, spectral theorem for normal matrices, Jordan normal form.
Intended learning outcomes
On completion of this subject, students should
Understand the concepts of:
- abstract groups, homomorphisms and quotient groups;
- abstract vector spaces, inner product spaces and linear transformations;
Be able to:
- do calculations in modular arithmetic and apply these to RSA cryptography;
- find eigenvalues, eigenvectors, minimal polynomials and normal forms for linear transformations;
- analyse groups of permutations, symmetries, and matrices;
- prove simple results in group theory and linear algebra.
In addition to learning specific skills that will assist students in their future careers in science, they will have the opportunity to develop generic skills that will assist them in any future career path. These include
- problem-solving skills: the ability to engage with unfamiliar problems and identify relevant solution strategies;
- analytical skills: the ability to construct and express logical arguments and to work in abstract or general terms to increase the clarity and efficiency of analysis;
- collaborative skills: the ability to work in a team;
- time management skills: the ability to meet regular deadlines while balancing competing commitments
Last updated: 3 March 2021