Introduction – Sets of Numbers (9/4)• Z - integers Z+ - positive integers• Q - rational numbers Q+ - positive rationals• R - real numbers R+ - 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.

• Zn - the set of numbers {0, 1, 2, ..., n – 1}

• U(n) - the subset of Zn consisting of numbers which are relatively prime to n.

• For example, what is U(12)? What is U(13)?

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.

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).

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?

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.• (Z12, +12) (i.e., Z12 under “addition mod 12”)

• (Z12*, .12) (i.e., Z12 under “multiplication mod 12”)

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