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.

3.1 Introduction to Groups

Abstract algebra is based on abstracting from algebra its core properties and studying algebra in terms of those properties. 41 Table 3.1. A closed binary operation ∗, defined on the set {a, b, c}. a b c a b c b b a c b c c b a

3.1.1 Binary Operations

We begin by extending the concept of addition. For a review of sets, see Sec- tion 2.1.1. Definition 3.1 binary operation A binary operation ∗ on a set S is a rule that assigns to each ordered pair a, b of elements of S some element in S. It must assign a single element to each pair otherwise it’s not defined or not well-defined , for assigning zero or more than one elements, respectively, and it must assign an element in S for the operation to be closed. If S is finite, we may display a binary operation ∗ in a table listing the elements of the set on the top and side of the table, and stating a ∗ b in row a, column b of the table, as in Table 3.1. Note that the operation defined by that table depends on the order of the pair, as a ∗ b = b ∗ a. Definition 3.2 commutative A binary operation ∗ on a set S is commutative if a ∗ b = b ∗ a for all a, b ∈ S. If S is finite, the table for a commutative binary operation is symmetric with respect to the diagonal from the upper-left to the lower-right. Definition 3.3 associative A binary operation ∗ on a set S is associative if a ∗ b ∗ c = a ∗ b ∗ c for all a, b, c ∈ S. If a binary operation ∗ is associative, we may write unambiguous long expres- sions without using parentheses.

3.1.2 Groups

The study of groups, as well as the need for new types of numbers, was moti- vated by solving equations.