Properties of Groups: Identity Element, Inverse, and Subgroups, Study notes of Algebra

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MT2505 Abstract Algebra

MRQ

March 17, 2021

Contents

Motivation: What is an algebraic

structure?

The purpose of this introductory chapter is to motivate the content of the module and specifically to consider the question “What is algebra?” To attempt to answer this ques- tion, we shall list various examples that will be typical algebraic structures. Algebra could then be viewed as the study of mathematical structures of the type we present here.

Perhaps previously you have seen the word “algebra” most commonly used in the context of polynomials. Indeed, you may have done a number of things with polynomials, for example:

• We might view a polynomial as a function f : R → R and then perform differentia- tion, integration and perhaps sketch the graph.
• We might solve polynomial equations. This could be achieved by factorizing the polynomial.
• We perform algebraic operations where we add and/or multiply polynomials.

This first of these tasks is not typical of the study of algebra. Although calculus can sometimes turn up in algebra, it is not central to the topic. The other two types of task, however, are directly related to the sort of thing that we do in algebra. Indeed, these two tasks are linked since to factorize a polynomial is to express it as a product of polynomials of smaller degree. A general description of what we study in algebra is algebraic operations such as the addition and multiplication that can be performed with polynomials. Other examples of structures where such algebraic operations occur include:

• The set of n × n matrices with entries that are real numbers. We can add and multiply such matrices to produce further matrices of the same size.
• The set TX of functions f : X → X from a fixed set X to itself. We can compose two such functions to produce another function in TX :

(f ◦ g)(x) = f (g(x)).

Composition is consequently an operation on the set TX of all functions X → X.

• We can add, multiply and subtract integers.
• We can add, multiply and subtract real numbers. We can divide two real numbers only when we divide by a non-zero real number.

Some of these examples might not seem the sort of thing that you were expecting to appear in the topic of “algebra” but they give some idea what the study of abstract algebra is concerned with. We wish to study structures that possess operations that behave a bit like addition or multiplication and then understand common themes that arise. By doing this in reasonable generality, the hope is that these common themes will become transparent (and this course — and those that follow — is intended to reveal these themes). We will finish this introductory chapter by making the core definition that will enable us to formulate the key ideas of the course.

Definition 0.1 Let A be a set. A binary operation on A is a function

A × A → A

defined on the set A × A = { (a, b) | a, b ∈ A } of pairs of points in A and that takes values in A.

Although we have defined a binary operation as a function, we will usually use a notation that suggests a similarity to addition or multiplication when denoting a binary operation. Thus, one example is that we might write a ∗ b to denote the image of a pair (a, b) under a binary operation. Other common notations are to use multiplicative notation where we write ab for the effect of combining a and b under the given binary operation and additive notation where we use the notation a + b. We shall view a binary operation as a way to combine two elements of the set A so as to result in another element of A (possibly one of the original two back again).

Example 0.2 The following defines a binary operation on the set R of real numbers:

a ∗ b = 0 for all a, b ∈ R.

This is a binary operation on R: It is not a very interesting binary operation.

We shall actually be interested in binary operations that have natural properties and which arise in interesting examples. We shall meet examples of such operations and their properties in our first main chapter.

Comments:

(i) The conditions listed above are often called the “axioms” of a ring. The labelling that we give for them is not universally adhered to. It has been chosen to coincide with that used in the module MT3505 Algebra: Rings and Fields. Note that A1–A are conditions relating only to the addition in a ring, while M2 refers only to the multiplication. Condition D is consequently significant as it says something about how addition and multiplication interact. If there were no condition involving both operations then this type of algebraic structure would not be so interesting: We would have two binary operations but they would not have any interaction with each other.

(ii) Note that Condition A3 ensures that every ring contains at least one element, namely the zero. Hence the underlying set R must be non-empty.

(iii) Usually we do not distinguish explicitly between an algebraic structure and its un- derlying set R. Thus we shall say

“R is a ring”

to mean: R is a set upon which addition and multiplication operations are defined such that Conditions A1–A4, M2 and D all hold.

(iv) Some textbooks (and some lecture notes) explicitly state two additional conditions

a + b ∈ R for all a, b ∈ R

and

ab ∈ R for all a, b ∈ R.

Note, however, that to say “+ is a binary operation” is to say that

R × R → R (a, b) 7 → a + b

is a function taking values in R. Thus the above two conditions are built into our requirement that addition and multiplication are binary operations on the given set R in Definition 1.1.

We have additional names that refer to the conditions appearing in Definition 1.1. These are the following terms:

Definition 1.2 Let A be any set and ∗ be a binary operation on A.

(i) We say that the binary operation ∗ is commutative if

a ∗ b = b ∗ a for all a, b ∈ A.

(ii) We say that the binary operation ∗ is associative if

(a ∗ b) ∗ c = a ∗ (b ∗ c) for all a, b, c ∈ A.

(iii) An identity for the binary operation ∗ is an element e ∈ A such that

e ∗ a = a ∗ e = a for all a ∈ A.

(iv) If there is an identity e for the binary operation ∗, then an inverse for a ∈ A is an element b ∈ A such that a ∗ b = b ∗ a = e.

Thus, a ring is a set endowed with two binary operations, addition and multiplication, satisfying the following properties:

• addition is commutative and associative,
• there is an additive identity (called zero),
• every element has an additive inverse (its negative),
• multiplication is associative,
• the distributive laws (D) hold.

How to verify that a mathematical object is a ring:

• Say what the underlying set R is.
• Say what the addition and multiplication operations are and verify these actually are binary operations: that is, check a + b ∈ R and ab ∈ R for all a, b ∈ R.
• Verify Conditions A1–A4, M2 and D. In the case of A3 and A4, this will usually involving stating explicitly what the zero and what the negatives are.

Example 1.3 The set Z of all integers forms a ring under the usual addition and multi- plication operations. The verification is as follows:

• The sum of two integers is always an integer and the product of two integers is again always an integer.
• Conditions A1, A2, M2 and D are standard properties of the usual arithmetic oper- ations.
• 0 ∈ Z is an integer and it satisfies the requirement that 0 + a = a + 0 = a for all a ∈ Z appearing in Condition A3.
• If a ∈ Z, then −a is also an integer and this satisfies a + (−a) = (−a) + a = 0, so Condition A4 holds.

Hence the set Z of integers is indeed a ring under the usual addition and multiplication operations.

This example illustrates how one verifies many easy examples are rings: namely those with which we are essentially familiar from the arithmetic we learnt to perform when we were much younger. In these cases, Conditions A1–A4, M2 and D are usually things that we have relied upon for years and so it feels perhaps unusual to actually make explicit reference to them. In the same vein, we record some other easy examples:

Matrix rings

Definition 1.6 Let R be any ring and n be a positive integer. An n × n matrix over R is an array consisting of n rows and n columns whose entries are selected from the ring R:

A =

a 11 a 12 · · · a 1 n a 21 a 22 · · · a 2 n .. .

an 1 an 2 · · · ann

where aij ∈ R for all i and j. We shall abbreviate this by writing A = [aij ] to indicate that the (i, j)th entry of A is the element aij. We define Mn(R) to be the set of all n × n matrices over R and define addition and multiplication of matrices as follows: If A = [aij ] and B = [bij ] are n × n matrices over R, then

A + B = [aij + bij ] AB = [cij ]

where

cij =

∑^ n

k=

aikbkj.

We call Mn(R) a matrix ring over R.

So to add two n × n matrices A and B, we simply add the corresponding entries of each matrix. To multiply A and B, the (i, j)th entry is equal to the sum of the values obtained when we multiply each entry of the ith row of A by the corresponding jth column of B. Both operations will be familiar to students from their previous studies (arising, for example, in MT1002 Mathematics and MT2501 Linear Mathematics). We have called Mn(R) a “matrix ring” and so one should actually verify that this is justified; that is, that the collection of n × n matrices with entries from R is indeed a ring in the sense of Definition 1.1. The full verification appears in these lecture notes, but some steps will be omitted in the lectures.

Theorem 1.7 Let R be a ring and n be a positive integer. Then the matrix ring Mn(R) is indeed a ring with respect to the addition and multiplication given in Definition 1.6.

Proof: Our definition of addition and multiplication of two n × n matrices involves adding and multiplying entries to insert into new matrices. The results are always matrices in Mn(R) and so addition and multiplication are binary operations on the set of n × n matrices over R. We must verify that the Conditions A1–A4, M2 and D in the definition of a ring. Let A = [aij ], B = [bij ] and C = [cij ] be arbitrary n × n matrices over R.

A1: By definition,

A + B = [aij + bij ] and B + A = [bij + aij ];

that is, the (i, j)th entry of A + B is aij + bij and that of B + A is bij + aij. However, here we are adding elements of the original ring R and we know that aij +bij = bij +aij since R itself satisfies Condition A1. Hence the entries of A + B and B + A are the same, so A + B = B + A.

A2: This is similar. We know that R is a ring so satisfies Condition A2, so a + (b + c) = (a + b) + c for all a, b, c ∈ R. We apply this in the following calculation to the (i, j)th entry of the matrix arising:

(A + B) + C =

[aij ] + [bij ]

• [cij ] = [aij + bij ] + [cij ] = [(aij + bij ) + cij ] = [aij + (bij + cij )] = [aij ] + [bij + cij ] = [aij ] +

[bij ] + [cij ]

= A + (B + C).

Hence Condition A2 holds in Mn(R).

A3: Let 0 denote the n × n matrix all of whose entries are 0 (the zero of the ring R). Since 0 + a = a + 0 = a for all a ∈ R, we now calculate

0 + A = 0 + [aij ] = [0 + aij ] = [aij ] = A

and similarly A + 0 = A. Hence 0 is a zero in Mn(R).

A4: Let us write −A for the matrix whose entries are the negatives of the entries of A; that is, −A = [−aij ]. Then

A + (−A) = [aij ] + [−aij ] = [aij + (−aij )] = [0] = 0.

Similarly (−A) + A = 0. This shows Condition A4 holds in Mn(R).

M2: We shall denote the (i, j)th entry of the product AB by (AB)ij. Recall this is given by the formula (AB)ij =

∑^ n

k=

aikbkj.

Similar formulae will be used for the (i, j)th entry of other products of matrices. Consequently, the (i, j)th entry of the product (AB)C is:

( (AB)C

ij =

∑^ n

k=

(AB)ikckj

∑^ n

k=

( (^) ∑n

`=

aibk

ckj

∑^ n

k=

∑^ n

`=

(aibk)ckj (since Condition D holds in the ring R)

∑^ n

k=

∑^ n

`=

ai(bkckj ) (since multiplication is associative in R (M2))

∑^ n

`=

∑^ n

k=

ai(bkckj )

Definition 1.8 A commutative ring R is a ring that satisfies the additional condition:

M1: ab = ba for all a, b ∈ R.

Example 1.9 (i) We know that multiplication in our familiar number systems is com- mutative. Thus, Z, Q, R and C are examples of commutative rings.

(ii) The above calculation shows that the matrix rings M 2 (Z), M 2 (Q), M 2 (R) and M 2 (C) are not commutative rings. The same idea shows that, for n > 2, none of the matrix rings Mn(Z), Mn(Q), Mn(R) and Mn(C) are commutative rings.

Polynomial rings

Definition 1.10 Let R be any ring. A polynomial over R is an expression of the form

f (X) = a 0 + a 1 X + a 2 X^2 + · · · + anXn

where n is a non-negative integer (n > 0) and a 0 , a 1 ,... , an ∈ R. The symbol X is called an indeterminate. If f (X) 6 = 0 (that is, not all coefficients are 0) and we choose the expression for f (X) so that an 6 = 0, then we say that f (X) has degree n.

We shall view two polynomials with indeterminate X as the same if they have the same coefficients, but one issue does arise. We wish to view the polynomials

a 0 + a 1 X + a 2 X^2 + · · · + anXn

and

a 0 + a 1 X + a 2 X^2 + · · · + anXn^ + 0Xn+1^ + · · · + 0XN

as being the same; that is, padding with lots of terms with 0 as coefficient should not make any difference. Consequently, suppose that

f (X) = a 0 + a 1 X + a 2 X^2 + · · · + anXn g(X) = b 0 + b 1 X + b 2 X^2 + · · · + bmXm

are two polynomials with indeterminate X and coefficients from the same ring R. Then we say f (X) and g(X) are equal (that is, f (X) = g(X)) if ak = bk for all k except that additionally one polynomial has further terms all with 0 as coefficient. This issue about padding using terms with 0 coefficients is one of the main compli- cations when working with polynomials. Allenby’s textbook [1] gets round this issue by treating polynomials as infinite sequences where from some point all subsequent entries are 0. (Thus he works with the sequence of coefficients in a polynomial, rather than the polynomial itself.) This is mathematically quite clean, but it feels we have moved some distance from the intuitive idea behind polynomials. Here we shall follow the usual route of working with polynomials as defined above as this fits with our already existing expe- rience. One needs to keep track of “padding with 0 coefficient terms,” but hopefully this will not prove to be too much of an obstacle for understanding. To simplify notation, we shall often write

f (X) =

akXk

to denote the polynomial f (X) = a 0 + a 1 X + · · · + anXn. This notation always denotes a finite sum with the understanding that any terms with 0 as coefficient can be inserted or omitted without changing the element. Having discussed what is meant by polynomials, let us now perform algebra with them.

Definition 1.11 Let R be a ring. The polynomial ring with indeterminate X over R is denoted by R[X] and is the set of all polynomials in X with coefficients in R and is endowed with the following binary operations: If f (X) =

akXk^ and g(X) =

bkXk, we define

Addition: f (X) + g(X) =

(ak + bk)Xk;

Multiplication: f (X) g(X) =

ckXk, where

ck = a 0 bk + a 1 bk− 1 + · · · + akb 0 =

∑^ k

i=

aibk−i.

The formula for addition of polynomials is relatively straightforward: we are simply adding the coefficients of corresponding terms in each polynomial. To motivate the formula for multiplication, observe that if we multiply two bracketed expressions

(a 0 + a 1 X + · · · + amXm)(b 0 + b 1 X + · · · + bnXn) (1.2)

then the coefficient in front of Xk^ in the product will be

ck = a 0 bk + a 1 bk− 1 + · · · + akb 0.

Note here that we are implicitly making use of the potential to pad the polynomials f (X) and g(X) with terms having 0 coefficient to make the above formulae make sense. For example, if f (X) has degree m and g(X) has degree n, then in the product in (1.2), the coefficient cm+n is actually given by

cm+n = ambn

since ai = 0 for i > m and bj = 0 for j > n in the product. We have called R[X] a “polynomial ring.” As before, one should therefore verify that it is indeed a ring. However, polynomials will arise only rarely in this module so although the full details appear in the lecture notes, they will be omitted in the lectures.

Theorem 1.12 Let R be a ring. Then the polynomial ring R[X] is indeed a ring with respect to the addition and multiplication given in Definition 1.11.

Proof: Our definition of addition and multiplication specifies the coefficients as being built from adding and multiplying coefficients from the ingredients. Thus our addition and multiplication are binary operations on the set of polynomials over R. We must verify that the Conditions A1–A4, M2 and D in the definition of a ring. Let f (X) =

akXk, g(X) =

bkXk^ and h(X) =

ckXk^ be arbitrary polynomials in the indeterminate X over the ring R.

A1: We use the fact that addition is commutative in the ring R: a + b = b + a for all a, b ∈ R. We apply this to the coefficients appearing in the following calculation:

f (X) + g(X) =

akXk^ +

bkXk

M2: To verify associativity of the multiplication in R[X] is a bit cumbersome. We can simplify the argument a little by observing that the formula for multiplication is

f (X)g(X) =

m

(i, j) with i+j=m

aibj

Xm;

that is, the Xm-coefficient is the sum of all products aibj where i + j = m. The calculation is then as follows: ( f (X)g(X)

h(X) =

i

aiXi

j

bj Xj

k

ckXk

m

(i, j) with i+j=m

aibj

Xm

k

ckXk

n

( (^) n ∑

m=

(i, j) with i+j=m

aibj

cn−m

Xn

n

(i, j, k) with i+j+k=n

aibj ck

Xn

(using associativity and distributivity of multiplication in R)

=

n

( (^) n ∑

i=

ai

(j, k) with j+k=n−i

bj ck

Xn

i

aiXi

m

(j, k) with j+k=m

bj ck

Xm

i

aiXi

j

bj Xj

k

ckXk

= f (X)

g(X)h(X)

This shows that Condition M2 holds in R[X]. D: Finally we demonstrate that the distributive law holds in R[X] (and this is more straightforward):

f (X)

g(X) + h(X)

i

aiXi

j

bj Xj

k

ckXk

i

aiXi

j

(bj + cj )Xj

i

( (^) ∑i

k=

ak(bi−k + ci−k)

Xi

i

( (^) ∑i

k=

(akbi−k + akci−k)

Xi

i

( (^) ∑i

k=

akbi−k

Xi^ +

i

( (^) ∑i

k=

akci−k

Xi

i

aiXi

j

bj Xj

i

aiXi

j

cj Xj

= f (X)g(X) + f (X)h(X).

This shows that the polynomial ring R[X] is indeed a ring, as claimed. 

As observed with matrix rings, it now follows that the sets of polynomials with integer, rational, real and complex coefficients (that is, Z[X], Q[X], R[X] and C[X]) are rings under the addition and multiplication we have defined.

Fields

We have commented earlier that division in the real numbers R can only be permitted when we do not attempt to divide by 0. Consequently, division is not a binary operation on R. There is a way to place division as a concept within the algebraic framework that we are presenting. This is usually done in the context of what is known as a field:

Definition 1.13 A field is a commutative ring F (that is, it has addition and multipli- cation defined upon it such that Conditions A1–A4, M1, M2 and D all hold) with two additional properties:

M3: there is an element 1 ∈ F with 1 6 = 0 such that 1a = a1 = a for all a ∈ F ;

M4: for every a ∈ F with a 6 = 0, there is some element a−^1 ∈ F such that aa−^1 = a−^1 a =

Thus a field is a commutative ring that has a multiplicative identity 1 with 1 6 = 0 and such that every non-zero element has a multiplicative inverse. The majority of students taking this module will have already met the term “field”, either having taken the module MT2501 Linear Mathematics or be taking that in parallel with this one.

Example 1.14 (i) From our familiarity with number systems, we know that Conditions M3 and M4 hold in the rational numbers Q, real numbers R and complex numbers C. Consequently, Q, R and C are our typical examples of fields.

(ii) If a is any integer, then 2a is an even number and so, in particular, 2a 6 = 1 for all a ∈ Z. Hence there is no multiplicative inverse for 2 in Z and we conclude that the ring of integers Z is not a field.

Subrings

This is only a brief introduction to the topic of rings. They are considered in much more detail in the Honours module MT3505 Algebra: Rings & Fields. The last thing we mention is the concept of a subring:

Definition 1.15 Let R be a ring and S be a subset of R that forms a ring under the same operation as defined on R. We then say that S is a subring of R.

Chapter 2

Greatest Common Divisors and

the Euclidean Algorithm

The material we present in this chapter concerns certain properties of the integers. It can, however, be placed in a more general setting: There is a special type of ring called a Euclidean domain where the same behaviour is seen. The ring of integers Z is an example of a Euclidean domain. The study of Euclidean domains is for another module (see MT3505 Algebra: Rings & Fields), but what we present here for integers will be needed later in this module. Some of this material also appears in the module MT Pure & Applied Mathematics, but since not everyone taking this module will have taken that module, we include full details here.

Definition 2.1 Let a, b ∈ Z.

(i) We say that a divides b if b = ac for some c ∈ Z. We also then say that a is a divisor of b. We write a | b to indicate that a divides b.

(ii) Suppose that at least one of a and b is non-zero. The greatest common divisor of a and b is the largest integer d which divides both a and b. We write gcd(a, b) to denote the greatest common divisor of a and b. (iii) If it is the case that gcd(a, b) = 1, then we say that a and b are coprime.

Comments:

(i) The symbol a | b is a statement about the way that a and b are related. It is not equal to a number and the “value” it takes is either “True” or “False.” In particular, note that a | b is not the same thing as a/b. The latter is a fraction, that is, an element of the rationals Q, while a | b is True when a is a divisor of b and is False when it is not.

(ii) Note that 0 = a × 0 for all integers a. Therefore every integer divides 0. This is why we assume that at least one of a and b is non-zero when defining the greatest common divisor since there is no largest divisor of 0. However, if a is any non-zero integer, then any divisor d of a satisfies d 6 |a|. Hence there is a greatest common divisor of a and b when a and b are not both zero.

(iii) The divisors of b and −b coincide: if b = ac, then −b = a(−c). Similarly if a divides b, then so does −a. In view of this, we shall work entirely with positive integers in this chapter.

(iv) For small integers, it is possible to calculate the greatest common divisor simply by listing the divisors of the two integers concerned and selecting the largest. For example,

gcd(2, 5) = 1 gcd(3, 9) = 3 gcd(15, 20) = 5.

(v) In particular, 2 and 5 are coprime integers. When a and b are coprime integers, it means that the only positive divisor of a and b is 1. We shall now describe a process to calculate the greatest common divisor of two positive integers a and b. It is useful particularly when a and b are relatively large and when we cannot immediately spot the divisors of the two integers. We specify the method as an algorithm: It is particularly suitable for implementation on a computer and is commonly found within computer algebra systems. Moreover, since gcd(a, b) = gcd(b, a), we can assume that a > b when applying our method.

Algorithm 2.2 (Euclidean Algorithm)

Input: Two integers a and b with a > b > 0.

Output: The greatest common divisor gcd(a, b).

Method:

• Step 1: Define a 1 = a and b 1 = b. Divide a 1 by b 1 to find quotient and remainder:

a 1 = b 1 q 1 + r 1 where 0 6 r 1 < b 1.

• Step n: Define an = bn− 1 and bn = rn− 1 arising from the previous step. Divide an by bn to find quotient and remainder:

an = bnqn + rn where 0 6 rn < bn.

• Repeat until rk = 0.
• The last non-zero remainder rk− 1 is gcd(a, b).

Example 2.3 Determine the greatest common divisor of 399 and 1554.

Solution: We apply the Euclidean Algorithm.

Step 1: Define a 1 = 1554 and b 1 = 399. Divide a 1 by b 1 :

1554 = 3 × 399 + 357

So r 1 = 357.

Step 2: Define a 2 = 399 and b 2 = 357. Divide:

399 = 1 × 357 + 42

So r 2 = 42.