Free Abelian Groups and Non-Commutative Groups - Algebraic Topology Review 4

Explore free abelian groups and non-commutative groups in this 51-minute lecture on algebraic topology. Learn about the importance of free abelian groups in algebraic topology, their modeling on the additive group of integers Z, and their similarity to vector spaces. Discover the Fundamental Theorem of Finitely Generated Commutative Groups, which describes the structure of these groups as a direct sum of integers and a finite commutative group. Delve into non-commutative groups, focusing on the example of S_3, the symmetric group on three objects. Examine the differences between left and right cosets in non-commutative groups, and understand the concept of normal subgroups and quotient groups. Gain valuable insights into these fundamental concepts of group theory and their applications in algebraic topology.