Abstract Algebra I - Auburn University

Abstract Algebra I

Randall R. Holmes Auburn University

Copyright c 2012 by Randall R. Holmes Last revision: November 15, 2018 This work is licensed under the Creative Commons AttributionNonCommercial-NoDerivatives 4.0 International License. To view a copy of this license, visit .

Notation

? N = {1, 2, 3 . . . }, natural numbers ? Z = {. . . , -2, -1, 0, 1, 2, . . . }, integers

m ? Q = | m, n Z, n = 0 , rational numbers (fractions)

n ? R, real numbers

? C = {a + bi | a, b R} (i = -1), complex numbers ? Q?, R?, C?, nonzero elements of Q, R, C, respectively ? Q+, R+, positive elements of Q, R, respectively ? Matm?n(R), m ? n matrices over R ? Matn(R), n ? n matrices over R ? GLn(R), invertible n ? n matrices over R ? SLn(R), n ? n matrices over R having determinant 1 ? FR, functions from R to R

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0 Introduction

There are many familiar ways in mathematics to combine two things to get another. Examples are

? addition of numbers, ? multiplication of numbers, ? addition of matrices, ? multiplication of matrices, ? addition of vectors, ? composition of functions.

There is some commonality among these operations. For instance, each is associative ((x + y) + z = x + (y + z), (xy)z = x(yz), etc.). We could define an "abstract associative structure" to be a set with an associative operation. Then we could study that abstract associative structure on its own knowing that anything we discovered would automatically apply to all of the examples above. This is the idea behind abstract algebra. The present course is the study of a "group," which is a set with an associative operation, having an identity element, and such that each element has an inverse (see Section 4). With some restrictions, each of the examples above gives rise to a group. Groups arise naturally in many areas of mathematics and in other areas as well (e.g., chemistry, physics, and electronic circuit theory).

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1 Function

1.1 Notation

Let X and Y be sets. A function f from X to Y (written f : X Y ) is a rule that assigns to each element x of X a unique element f (x) of Y , depicted using a Venn diagram like this:

X

f

Y

x

f (x)

We think of x as an input to the function f and f (x) as the corresponding output. The set X is the domain of f and the set Y is the codomain of f. A function is sometimes described by giving a formula for the output in terms of the input. For instance, one writes f (x) = x2 to refer to the function f : R R that takes an input x and returns the output x2. So the equation f (x) = x2 says that for an input x "the output f (x) is x2."

1.2 Well-defined function

For a proposed function to be well defined (and hence actually be a function) it must assign to each element of its domain a unique element of its codomain. The following example shows various ways a proposed function can fail to be well defined.

1.2.1 Example The following proposed functions are not well defined (and are therefore not actually functions):

(a) f : R R given by f (x) = 1/x. (The number 0 is in the domain R, but f (0) = 1/0 is undefined, so f does not assign an element to each element of its domain.)

(b) g : [0, ) R given by g(x) = y where y2 = x. (We have g(4) = 2

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(since 22 = 4), but also g(4) = -2 (since (-2)2 = 4), so g does not assign a unique element to each element of its domain.)

(c) h : [2, ) (4, ) given by h(x) = x2. (The number 2 is in the domain [2, ), but h(2) = 4, which is not in the codomain, so h does not assign to each element of its domain an element of its codomain.)

1.3 Equal functions

Two functions f and g are equal (written f = g) if they have the same domain and the same codomain and if f (x) = g(x) for each element x in their common domain.

1.3.1 Example Let f : (-, 0] R be given by f (x) = -x and let g : (-, 0] R be given by g(x) = x2. Prove that f = g.

Solution For every x (-, 0], we have

f (x) = -x = |x| = x2 = g(x). Therefore, f = g.

1.4 Injective function

Let f : X Y be a function. Informally, f is "injective" if it never sends two inputs to the same output:

X

f

Y

X

f

Y

f injective Here is the formal definition:

3

f not injective

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