Welcome to our activities for Week 10 (the week of November 30).
Our theme 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.
In our first video lecture we define a homomorphism and give some examples.
In our second video we look at the kernel and image of a group homomorphism, which are subgroups of the domain and the codomain groups respectively. We consider the question of how the image of a homomorphism can be constructed from the domain and the kernel. This will lead us to the concept of quotient groups, which is next week's theme and our last topic for this module.
This week's content is in Section 4.1 of the lecture notes, which as usual contains some extra detail and additional examples to the ones in the videos.