Categorical Perspective: Group Quotients

This lecture explores the categorical sense of constructing a group quotient by a normal subgroup, showing it as a specific example of a more general construction called the push-out. The push-out is defined by a unique homomorphism that satisfies a universal property, where any pair of homomorphisms commute. The lecture delves into the terminology related to push-outs and the unique homomorphisms involved. It also discusses the push-out of groups, the relationship between normal subgroups, and the universal property of finding a homomorphism. The instructor emphasizes the importance of showing the smallest normal subgroup and the goal of finding a homomorphism that satisfies specific conditions.