Alpha Complex Alpha Shapes
3
Group Theory
Having examined the structure of the input to our computations in the last chapter, we now turn to developing the machinery we need for characterizing
the topology of spaces. Recall that we are interested in classification systems. Group theory
provides us with powerful tools to define equivalence relations using homomorphisms and factor groups. In the next chapter, we shall utilize
these tools to define homology, a topological classification system. Unlike homeomorphy and homotopy, homology is discrete by nature. As such, it is
the basis for my work.
The rest of this chapter is organized as follows. In Section 3.1, I will intro- duce groups. I devote Section 3.2 to developing techniques for characterizing
a specific type of groups: finitely generated Abelian groups. In Section 3.3, I examine advanced algebraic structures in order to generalize the result from
the previous section.
Abstract algebra is beautifully lucid by its axiomatic nature, capturing fa- miliar concepts from arithmetic. The plethora of arcane terms, however, often
makes the field inscrutable to nonspecialists. My goal is to make the subject thoroughly accessible by not leaving anything obscure. Consequently, there is
a lot of ground to cover in this chapter. My treatment is derived mostly from the excellent introductory book on abstract algebra by Fraleigh 1989, which
also contains the proofs to most of the theorems stated in this chapter. I used Dummit and Foote 1999 for the advanced topics.