Push-out and Quotients

This lecture explores the concept of push-out and quotients in group theory from a categorical perspective, focusing on homomorphisms, normal subgroups, and the unique homomorphisms between groups. The instructor presents Proposition 2.5, which establishes the push-out of a group homomorphism. Various proofs are provided to demonstrate the properties of push-outs and their relation to normal subgroups. The lecture concludes with a discussion on the categorical study of quotients and previews the upcoming exploration of isomorphism theorems.