Cayley’s Theorem and Group Homomorphisms

Welcome to our activities for Week 9 (the week of November 1).

This week we continue with the topic of group actions. Our first topic is Cayley’s Theorem, another famous theorem about finite groups. A slightly vague way to describe it is to say that every group of order n can be imagined as a subgroup of the symmetric group S_n. “Can be imagined” is not a mathematically precise term obviously, but we will tidy that up.

Our second topic this week is group homomorphisms. These are functions between groups, that are compatible with the group structure in both the domain and codomain (target) groups. Functions are important in every area of mathematics, but in most areas of mathematics we care not necessarily about all functions, but about the ones that make sense in the context of the themes of the subject. So in calculus we don’t study all functions on the set of real numbers, but only those that have some property like continuity or differentiability, that make them amenable to the objects of calculus. In linear algebra, we are interested not in all functions between vector spaces, but in linear transformations. They are the functions that respect the linear structure and can be represented by matrices. The analogous concept for functions between groups is a group homomorphism.

The relevant sections of the lecture notes this week are Section 3.3 and Section 4.1

Slides for this week’s lectures.

We have three (old) videos this week, the first one is about the Orbit-Stabilizer Theorem, which was in last week’s lectures.