Quotient Groups and Normal Subgroups (Algebra 1: Lecture 4 Video 4)

Lecture 4: This lecture focused on understanding quotient groups. We first talked about the group structure on the set of fibers of a group homomorphism. We then discussed the relationship between cosets and fibers. We saw that in the case where our subgroup is given by the kernel of a homomorphism, we could put a group structure on the set of left cosets. The most important thing here was showing that the operation we defined was 'well-defined', that it does not depend on a choice of coset representatives. We then explained how this operation led to a group structure on the set of left cosets of a subgroup N in G if and only if N is a normal subgroup of G.

Reading: The material of this lecture comes from Section 3.1. We will start the next lecture by talking about Proposition 7 of this section. We will then move on to Section 3.2.