Group Homomorphisms

This lecture explores the concept of group homomorphisms, focusing on constructing homomorphisms between groups using generators and relations. The instructor demonstrates how to build an isomorphism between a free group and a dihedral group, emphasizing the importance of choosing the images of the generators. The lecture also delves into the properties of normal subgroups and the process of verifying normality within a group. Through detailed examples and calculations, the instructor illustrates how to determine the order of a group and establish isomorphisms between different groups. The lecture concludes by discussing the construction of groups with specific properties, such as the quaternion group and the octonion algebra, showcasing the application of group theory in various mathematical contexts.