Groups and Homomorphisms

This lecture introduces the concept of Euler's totient function and delves into the definition of a group, exploring examples such as rotations around the origin and the additive group of integers modulo n. The lecture covers the order of a group, homomorphisms, isomorphisms, and the cyclic group of order n. It also discusses the kernel of a homomorphism and provides examples of group homomorphisms, including a detailed explanation of the kernel. The lecture concludes with a discussion on generators and relations in groups, presenting the notion of a group presentation and illustrating the construction of a group isomorphism between cyclic groups. The importance of group theory in mathematics and its applications in various branches are highlighted.