To introduce groups and rings, driven by both examples and elementary theory.
Group Theory: motivating examples (numbers, cyclic group, dihedral group, symmetric group, transformations of the plane), elementary properties, subgroups, odd and even permutations.
Ring Theory: commutative and non-commutative rings, fields, examples (the integers, polynomials with integer coefficients, polynomials with real coefficients and quotient rings of such, unit groups, factorisations in the integers and polynomials.
List of covered algebraic definitions: group, subgroup, group homomorphism (including kernel, image, isomorphism), order, sign of permutation, ring, field, subring, ideal, ring homomorphism (including kernel, image, isomorphism), quotient ring
To explore a wide range of examples of groups and rings; to cover elementary properties of both structures; to start to look at ways to study and classify groups and rings as abstract objects. As a result of taking this module, students should be sufficiently prepared for the appearance of groups and rings in other first year courses and in follow-up second year courses.
Any book with ‘Abstract Algebra’ in the title is worth looking at (there are many). Here are a few specific recommendations (but note that comprehensive lecture notes will be provided):
Lara Alcock – How to Think About Abstract Algebra
Tony Barnard and Hugh Neill - Discovering Group Theory
I.N.Herstein - Topics in Algebra
Nathan Jacobson - Lectures in Abstract Algebra