1. Introduction – Sets of Numbers (9/4) Z - integers Z + - positive integers Q - rational numbersQ + - positive rationals R - real numbersR + - positive reals C - complex numbers For any number set S, by S* we mean the set with 0 removed. So, for example, Q* means all non-zero rationals. Z n - the set of numbers {0, 1, 2,..., n – 1} U(n) - the subset of Z n consisting of numbers which are relatively prime to n. For example, what is U(12)? What is U(13)?

2. Algebraic Objects Any set which has one or more binary operations on it is called an algebraic object. (A binary operation on a set combines two elements of the set to produce a third element of the set. For example, R has 4 binary operations but Z has only 3. What are they and why?) Abstract Algebra is the study of algebraic objects, both from a general, abstract point of view and from looking at many examples. There are many types of abstract algebraic objects: groups, rings, fields, vector spaces, modules, etc. In this course, we concentrate on groups since in some ways they are the simplest.

3. Loose Definition of a Group We will be somewhat more precise shortly, but for the moment we consider the following definition: A group (G,  ) is a set G possessing a single binary operation  such that: (Existence of an identity element) There exists an element e in G such that for every a  G, a  e = e  a = a. (Existence of inverses) For every element a  G, there exists an element a -1  G such that a  a -1 = a -1  a = e. When working with an abstract group G, we often omit the symbol  and simply use “juxtaposition” (i.e., write a b in place of a  b).

4. Simple Example of a Group Consider the set Z and operation +. Is + a binary operation on Z? Does there exist an identity element for + in Z? If so, what is it? Given in element a  Z (i.e., given any integer), is there another element a -1  Z such that a + a -1 = the identity element? If so, what is it? So, is (Z, +) a group?

5. Is it a group? Yes (A) or No (B) (Z +, +) (i.e., positive integers under addition) (Z,. ) (i.e., the integers under multiplication) (2Z, +) (i.e., the even integers under addition) (Q, +) (Q,. ) (Q*,. ) (R[x], +) (i.e., all polynomials with real coefficients under +) All 2 by 2 matrices with coefficients in Q under matrix multiplication. (Z 12, + 12 ) (i.e., Z 12 under “addition mod 12”) (Z 12 *,. 12 ) (i.e., Z 12 under “multiplication mod 12”)

6. Assignment for Friday Obtain the text. Do the follow-up assignment (not to hand in).