4 Group Theory
Afra Zomorodian · 2004
Last lecture, we learned about a combinatorial method for characterizing spaces: using simplicial complexes as triangulations of the spaces. We also showed how the connectivity of compact 2-manifolds is fully characterized by an invariant, the Euler characteristic, that may be computed using any triangulation of the space. In this lecture, we study group theory. This theory is part of the beautiful machinery of abstract algebra, which is based on abstracting from algebra its core properties, and studying algebra in terms of those properties. Because of this abstraction, group theory is fundamental and applicable to questions in many theoretical fields such as quantum physics and crystallography, as well as questions in practical fields, such as establishing bar codes for products, serial numbers on currency, or solving Rubik’s Magic cube. For us, the theory provides powerful tools to define equivalence relations using homomorphisms and factor groups. These equivalence relations will enable us to partition the space of manifolds into coarser classifications that are computable. We begin with an introduction to groups, their subgroups, and associated cosets. We then look at how we factor a group much like the way we divide a composite integer. We end by developing techniques for characterizing a specific type of groups: finitely generated abelian groups.