# Part IB - Groups, Rings and Modules

## Lectured by O. Randal-Williams, Lent 2016

These notes are not endorsed by the lecturers, and I have modified them (often significantly) after lectures. They are nowhere near accurate representations of what was actually lectured, and in particular, all errors are almost surely mine.

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# Contents

- V Full version
- 0 Introduction
- 1 Groups
- 1.1 Basic concepts
- 1.2 Normal subgroups, quotients, homomorphisms, isomorphisms
- 1.3 Actions of permutations
- 1.4 Conjugacy, centralizers and normalizers
- 1.5 Finite p-groups
- 1.6 Finite abelian groups
- 1.7 Sylow theorems
- 2 Rings
- 2.1 Definitions and examples
- 2.2 Homomorphisms, ideals, quotients and isomorphisms
- 2.3 Integral domains, field of factions, maximal and prime ideals
- 2.4 Factorization in integral domains
- 2.5 Factorization in polynomial rings
- 2.6 Gaussian integers
- 2.7 Algebraic integers
- 2.8 Noetherian rings
- 3 Modules