Abstract Algebra – Theory and Applications

Preliminaries

A certain amount of mathematical maturity is necessary to find and study applications of abstract algebra. A basic knowledge of set theory, mathematical induction, equivalence relations, and matrices is a must. Even more important is the ability to read and understand mathematical proofs. In this chapter we will outline the background needed for a course in abstract algebra.

1.1 A Short Note on Proofs 

Abstract mathematics is different from other sciences. In laboratory sciences such as chemistry and physics, scientists perform experiments to discover new principles and verify theories. Although mathematics is often motivated by physical experimentation or by computer simulations, it is made rigorous through the use of logical arguments. In studying abstract mathematics, we take what is called an axiomatic approach; that is, we take a collection of objects S and assume some rules about their structure. These rules are called axioms. Using the axioms for S, we wish to derive other information about S by using logical arguments. We require that our axioms be consistent; that is, they should not contradict one another. We also demand that there not be too many axioms. If a system of axioms is too restrictive, there will be few examples of the mathematical structure.

A mathematical proof is nothing more than a convincing argument about the accuracy of a statement. Such an argument should contain enough detail to convince the audience; for instance, we can see that the statement “2x = 6 exactly when x = 4” is false by evaluating 2 · 4 and noting that 6 6= 8, an argument that would satisfy anyone. Of course, audiences may vary widely: proofs can be addressed to another student, to a professor, or to the reader of a text. If more detail than needed is presented in the proof, then the explanation will be either long-winded or poorly written. If too much detail is omitted, then the proof may not be convincing. Again it is important to keep the audience in mind. High school students require much more detail than do graduate students. A good rule of thumb for an argument in an introductory abstract algebra course is that it should be written to convince one’s peers, whether those peers be other students or other readers of the text.

Let us examine different types of statements. A statement could be as simple as “10/5 = 2;” however, mathematicians are usually interested in more complex statements such as “If p, then q,” where p and q are both statements. If certain statements are known or assumed to be true, we wish to know what we can say about other statements. Here p is called the hypothesis and q is known as the conclusion.