This module is not currently running in 2024 to 2025.
Groups are sets with a single binary operation. They arise as symmetry groups in contexts from puzzles like Rubik's cube to chemistry, where they help list molecules with a given number of atoms involved. In contrast, rings have two binary operations, generalising the arithmetic of integer numbers. This part of algebra has many applications in electronic communication, in particular in coding theory and cryptography. Outline Syllabus includes: permutations and cycle decomposition, subgroups, cosets, Lagrange's theorem, normal subgroups, symmetry groups, group actions, homomorphisms of groups and rings, ideals, factorization in rings, polynomial rings, domains, fields, quotient fields, finite fields.
48
80% Examination, 20% Coursework
See the library reading list for this module (Canterbury)
On successful completion of this module students will be able to:
a) cite and understand a representative selection of the definitions and terms of basic Group Theory and Ring Theory;
b) cite examples of the main mathematical structures introduced including groups, subgroups, quotient groups, rings, subrings, ideals and homomorphisms;
c) execute simple proofs and deductions from the axioms for these structures and express the reasoning with reasonable clarity;
d) perform simple calculations for specific examples of these structures;
e) appreciate the relevance of abstract algebraic structures to related areas of mathematics
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