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BSc Computer Science and Mathematics / Course details
Year of entry: 2024
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Course unit details:
Groups and Geometry
|Available as a free choice unit?
Groups are abstract algebraic structures which are central to almost every area of modern pure mathematics, and also to many application areas. In geometry, they manifest themselves as the natural way to describe and study symmetry. The course will provide an introduction to the algebraic theory of groups, and to some of their applications in geometric settings. It will also develop skills in abstract reasoning and mathematical writing, which are essential for later study in pure mathematics and also applicable in other areas.
|Mathematical Foundation & Analysis
The unit aims (i) to introduce students to fundamental notions of abstract algebra which are central to most of pure mathematics and many applications, (ii) to illustrate how these ideas can be applied in a geometric setting and (iii) to develop the skills students will need for later study in (especially pure) mathematics, such as reasoning about mathematical objects and accurate writing of mathematics proofs and arguments
- Reason accurately about abstractly defined mathematical objects, constructing formal arguments to prove or disprove mathematical statements about the objects introduced in this course and prerequisite courses, and distinguishing between correct and incorrect reasoning.
- Write mathematics (including proofs) accurately and clearly, making appropriate use of both the English language and mathematical notation.
- Recognise how abstract structures such as groups can manifest themselves in different settings, and apply knowledge about abstract groups to solve problems in concrete settings where they arise.
- State the group axioms, identify and calculate with common examples of groups, and determine whether or not a given structure is a group.
- Define, recognise and reason about basic concepts of group theory (such as groups, subgroups, cosets, conjugacy, homomorphisms, isomorphisms, factor groups, group actions, orbits, stabilisers, fundamental sets/domains and quotient spaces).
- State, apply and recall the proofs of some elementary theorems of group theory (such as Lagrange’s Theorem, the First Isomorphism Theorem and the Class Equation) and apply this knowledge in familiar and unseen settings.
- Define basic geometric concepts of Euclidean space and of the Riemann sphere, and solve elementary problems involving these.
- Determine which elements are conjugate in a given group using classification theorems, e.g. for symmetric groups, isometries of Euclidean space, or Möbius transformations. Determine explicit conjugators/centralisers.
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