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syllabus

syllabus

rref

efer

erence

ence

Core topic:

Introduction to groups

In this

In this

cha

chapter

pter

4A Groups

4B

The terminology of groups

4C Properties of groups

4D Further examples of groups

— transformations

4

An introduction

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M a t h s Q u e s t M a t h s C Y e a r 1 1 f o r Q u e e n s l a n d

Groups

Up until now, when you used the term algebra you probably thought of variables and operations with those variables. However, algebra exists in a wide variety of forms, from Boolean algebra to group algebra. In its most simple form it can be thought of as the science of equations.

At this stage in your mathematics studies you would be familiar with the use of classical algebra to solve equations such as x2− 9 = 0, yielding the solutions of x= ±3. However, in the 19th century, mathematicians gradually realised that mathematical symbols did not necessarily have to stand for numbers, if anything at all! From this idea, modern or abstract algebra arose.

Modern algebra has two main uses:

1. to describe patterns or symmetries that occur in nature and mathematics, such as different crystal formations of certain chemical substances 2. to extend the common number systems to

other systems.

In algebra, symbols that can be lated are elements of some set and the manipu-lation is done by performing certain operations on elements of that set. The set involved is referred to as an algebraic structure. The symbols may represent the symmetries of an object, the position

of a switch, an instruction to a machine or design of a statistical experiment. These symbols may then be manipulated using the familiar rules used with numbers.

A group is a system of elements with a composition satisfying certain laws.

It is hoped that this brief introduction to groups expands your understanding of the versatile and all-encompassing concept of algebra. The following section of work fits into this field of study by virtue of the fact that it deals with symbols and operations.

But first a new tool to help you deal with some notions used in groups.

Modulo arithmetic

Not to be confused with the modulus of a number (see chapter 1 on real numbers, R, where the modulus of −4, written | −4| = 4), modulo arithmetic uses a finite number system with a finite number of elements. This is sometimes referred to as ‘clock arithmetic’ because of the similarities with reading the time on an analog clock.

Algebraic structures

Research the topic of algebraic structures examining early algebraic systems that developed in ancient civilisations such as the Indian, Arabic, Babylonian, Egyptian and Greek. Highlight differences and similarities among the various forms.

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167

Consider reading the time shown on the clock face to the right. Whether it is 2 am or 2 pm we would say it is 2 o’clock, but in 24-hour time the 2 pm would be 1400 hours. In effect we have subtracted 12 hours from the 1400 (14 hours) to give an answer of 2. In this case we say that 2 is the residue, or what is left over when 12 hours is subtracted from the 14.

In modulo 12 arithmetic the same principle is used except that the 12 is replaced by a 0.

5 + 6 = 11 5 + 7 = 0

5 + 8 = 1 and so on.

In our normal decimal system 5 + 8 = 13, but in modulo 12 arithmetic the residue of 1 differs from 13 by 12 (or a multiple of 12) and 1 and 13 are said to be congruent. That is, in modulo 5 arithmetic, the numbers 3, 8 and 13 are congruent and in modulo 12 arithmetic, 2, 14, and 26 are congruent numbers.

Using more precise terminology, addition modulo 10 is written 3 + 9 ≡ 2 mod 10, 5 + 5 ≡ 0 mod 10, and so on.

(Note the abbreviation of modulo to mod.)

In mod 12, the numbers 0 to 11 are referred to as residues, as with 0 to 5 in mod 6. This information can be stored in a table, known as a Cayley Table.

12 1 2 4 5 7 8 10 11 6 3 9 0 1 2 4 5 7 8 10 11 6 3 9

Draw up a Cayley Table that shows the residues using addition modulo 4.

THINK WRITE/DRAW

Draw an empty table with 0, 1, 2, 3 in the first row and column and put a + sign in the top corner.

Start working across the first row. 0 + 0 = 0 etc. and do likewise with the first column.

The residues are the numbers left over when 4 is taken from the answer (if the answer is 4 or greater). As you

complete the table note that the answers are less than 4. So, for 2 + 2 the residue is 0.

1 + 0 1 2 3

0 1 2 3

2 + 0 1 2 3

0 1 2 3 0 1 2 3

1 2 3

3 + 0 1 2 3

0 1 2 3 0 1 2 3 1 2 3 0 2 3 0 1 3 0 1 2

1

WORKED

Example

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M a t h s Q u e s t M a t h s C Y e a r 1 1 f o r Q u e e n s l a n d

Groups

1 List 4 numbers congruent to:

a 4 in mod 8

b 4 in mod 6.

2 List the residues in:

a mod 3 b mod 9 c mod 11.

3 Draw up a Cayley Table that shows the residues for each of the following:

a addition mod 6

b multiplication mod 4

c multiplication mod 5.

The terminology of groups

Mathematics is often referred to as a science — sometimes as the science of patterns. You will enjoy your studies of mathematics more if you look for patterns in all the ideas you explore. One such idea is that of groups.

Group theory is applied to many areas of science such as genetics, quantum theory, molecular orbits, crystallography and the theory of relativity. In mathematics, group theory is applied to many mathematical models involving algebra, number theory and geometry. In chapter 1 you dealt with different sets of numbers within the Real Number System. Throughout your student life you have used the operations of addition, multi-plication, subtraction and division, finding a square root, reciprocals, and so on. These are examples of operations performed on numbers that are part of a certain set.

Operations (such as addition) that involve 2 input values, for example 2 + 3, are

called binary operations. Those that involve only one input value, such as finding the

square root of a number (for example ) are called unary operations. Others that

involve 3 input values are called ternary; for example, the principal, interest and term

of a loan are the 3 input values involved in calculating the amount of interest due on a loan. (Strictly speaking the multiplication involved is still carried out on pairs of values.)

Definition of terms

Groups that we will deal with consist of a system that involves a set of elements (often

numbers) and a binary operation. Lower case letters: a, b, c … are used to refer to

elements of the set and the symbol ‘

°

’ denotes whatever operation is involved.

For a non-empty set of elements S = {a, b, c, …} involved in the binary operation ‘

°

to be a group, G = [S,

°

], the following properties must hold.

remember

1. Modulo arithmetic is like clock arithmetic where 5 + 9 ≡ 4 in mod 10. 2. The residues of modulo x are all the whole numbers less than x. 3. Congruent numbers in mod x all differ by multiples of x.

remember

4A

SkillS HEET

4.1

WORKED

Example 1

8

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C h a p t e r 4 A n i n t r o d u c t i o n t o g r o u p s

169

1 Closure

An operation is closed if the result of that operation is an element of the same set as the 2 inputs.

For example 2 + 3 = 5 could be written 2

°

3 where S = {Real numbers} (or R) and

°

is the operation of addition.

This operation is closed because 5 ∈R

But consider 2 − 3 = −1 where S = {Natural numbers} (or N) and

°

is the operation of

subtraction. Because the result (−1) is not a member of the set of natural numbers this

operation is not closed. That is, the answer is not part of the initial set of natural numbers.

2 Associativity

If an operation is associative, the order in which operations are performed does not affect the answer.

Often brackets are employed to determine the order of operations. For example, consider (2 × 3) × 4 and 2 × (3 × 4)

(2 × 3) × 4= 6 × 4 2 × (3 × 4) = 2 × 12

= 24 = 24

In this case, both answers are the same. Note that only the position of the brackets changes and the order of the numbers remains the same.

But consider the operation of division:

(20 ÷ 2) ÷ 4 and 20 ÷ (2 ÷ 4)

= 10 ÷ 4 = 20 ÷ 0.5

= 2.5 = 40

Here the answers are not the same.

Division, like subtraction, is not associative. You would have realised this in your earlier junior mathematics studies.

3 Identity

For all elements of a set, if a unique element exists in the set such that a

°

u=a

then u is the identity element (IE) for that operation.

That means that there is only one element that leaves every element unchanged when the operation ‘

°

’ has been applied.

For example, 3 + 0 = 3 then 0 is the identity element for addition (IE+) for real

numbers.

However, 3 × 0 = 0 so 0 is not the identity element for real numbers under the operation of multiplication.

Note: The one identity element must work for all elements of the set so 5 + 0 = 5

and −8 + 0 =−8.

4 Inverse

For each element of a set there is a unique element a–1 such that a

°

a–1=u where

u is the identity element for that operation.

Unique means that every element has only one inverse.

2 × = 1 where 1 is the identity element for multiplication (IE×) Therefore is the multiplicative inverse of 2.

Now consider 2 +−2 = 0 where 0 is IE+; in this case −2 is the additive inverse of 2. However, note that the set involved here would have to be integers (that is, both

posi-tive and negaposi-tive) not just whole numbers because −2 ∉ {Whole numbers}.

1 2

---1 2

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We can now restate the definition of a group.

If the following 4 properties hold for a set of elements under a certain operation ‘

°

’: 1. closure

2. associativity

3. existence of an identity element 4. existence of an inverse

then the system under investigation [S,

°

] is a group. If a fifth property, commutativity, also holds, then the group is an Abelian group.

Commutativity

If the order of the elements involved has no effect on the outcome, then the operation is commutative.

For example 2 × 5 = 10 and 5 × 2 = 10

Hence multiplication with real numbers is commutative. Note the stated condition, ‘with real numbers’ because you have already worked with matrices where multi-plication in not commutative.

However, consider 10 ÷ 2 = 5 and 2 ÷ 10 = 0.2

So division is not commutative. You would be familiar with other operations as well that are not commutative.

Find a the identity element and b inverse for the operation defined as a

°

b = a + b + 2.

THINK WRITE

a An identity element is an element that, when involved in an operation with another element does not change the value of that element.

a Let a + b+ 2 =a (where b= IE)

therefore b=−2

IE =−2

b An inverse is an element that, when involved in an operation with another element results in the IE for that operation.

b Therefore a + b+ 2 =−2

(where b is the inverse of a and –2 = IE from part a)

a + b =−4 b=−4

2

WORKED

E

xample

Find the identity element for the operation defined as = a.

THINK WRITE

An IE is an element that, when involved in an operation with another element does not change the value of that element.

=a let b= IE Square both sides:

a2+b2=a2

therefore b2= 0

and b= 0 therefore IE = 0

a2+b2

a2+b2

3

WORKED

E

xample

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171

The terminology of groups

1 a Show that a

°

b= is not closed with respect to whole numbers.

b State its identity element.

2 If an operation a

°

b is defined as determine whether this is closed if a and b

are whole numbers. (Remember you only have to find one example where the operation is not closed to disprove a statement.)

History

of mathematics

N I E L S H E N R I K A B E L ( 1 8 0 2 – 1 8 2 9 )

During his life . . . Lord Byron, the

English poet, writes Don Juan.

Napoleon Bonaparte becomes emperor of France. Jean-Baptiste Lamarck, the French biologist, proposes that acquired traits are inherited by indi-viduals in a population.

Niels Abel was one of the most productive mathematicians of the 19th century. Born in Norway on 5 August 1802, by the age of 16 he had started his private study of the mathematics

of Newton, Euler, Gauss and Lagrange. As the sole supporting male of his family at 18 he tutored private pupils while continuing his own mathematical research. By the age of 19 he had proved that there was no finite formula for the solution of the general fifth degree polynomial.

He died of tuberculosis on 6 April 1829, two days before the announcement of his posting as professor to the Berlin university. His life in poverty stands in contrast to the regard with which he is held in his field; the term

Abe-lian group being named in honour of Abel. His

studies on group theory were central to the development of abstract algebra.

Questions:

1. How did Abel financially support his family?

2. Which property do groups bearing his name exhibit?

remember

A set S forms a group under the operation ‘

°

’ if and only if (iff) all of the following are true:

1. it is closed under ‘

°

’; that is, the result is an element of S

2. the order in which operations are performed has no effect on the results; that is, it is associative

3. there is only 1 identity element (IE), u, such that a

°

u = a

4. there is a unique inverse a1 for every element such that a

°

a–1= u, where u = IE.

5. If the property of commutativity also holds, then it is an Abelian group.

remember

4B

a+b

2

---a2+b2

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3 Find a the identity element and b the inverse for the operation

°

on real numbers

where a

°

b=a+b− 1.

4 What is the identity element of the operation a

°

b =a +bab if a and b are real numbers?

5 The operation a

°

b = 4ab2 is defined for positive real numbers a and b. Find the identity element for this operation.

6 Develop a proof to show that a

°

b= has no identity.

7 An operation is defined with respect to an ordered pair of integers as

(a, b)

°

(c, d) = (ad+ bc, bd). Show that (0, 1) is the identity element for the operation.

8 Show that a

°

b= (a+b)2 has no identity for real numbers.

Properties of groups

WORKED Example 2

WORKED Example 3

a+b

ab

---a Verify that the set of integers forms a group under addition. b Is this group Abelian?

THINK WRITE

a What numbers are involved? All

positive and negative integers and 0 are involved so state the set and operation. While you can think of actual values for the integers

(−1, 0 4 …) your answer should use

only variables, with constants used as examples only.

a Let Z = {a, b, c, …} be the set of integers; the operation is addition.

Test each of the 4 properties in the same order each time to help you remember the 4 tests.

iii The sum of any 2 integers is an

integer.

iii The order in which the operation is performed has no effect on the result.

iii Since 0 ∈ Z, IE+ exists.

iv Since Z contains all positive and

negative whole numbers, the inverse is −a

iii The operation is closed:

a+b=c where a, b and c∈ Z

iii The operation is associative: (a+b) +c=a+ (b+c) iii The identity element exists:

a+ 0 =a

iv The inverse exists:

a +−a= 0 State that the system forms a group

under the conditions stated.

Thus the set of integers forms a group under addition.

1

2

3

4

WORKED

E

xample

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173

Note that the test for commutativity is performed last because the first 4 properties are necessary to state that it is a group in the first place, before it is shown to be Abelian. This group, G = [Z, +] is an infinite group, having an unlimited set of elements. You will also deal with finite groups which have a countable number of elements.

THINK WRITE

b If the group is Abelian we need to show that this operation is commutative.

b Commutativity

a+b=b+a

Therefore the group is Abelian.

Verify that the set of odd integers does not form a group under addition.

THINK WRITE

What numbers are involved? The set of odd integers includes

−5, −3, −1, 1, 3, 5 … State the set and operation.

S = {a, b, c, …} is the set of odd integers. The operation is addition.

Test the 4 properties as shown in worked example 4.

Closure: a+b∈ S Let a= 3 and b= 5 3 + 5 = 8 and 8 ∉ S Therefore G ≠ [S, +]

There is no need to proceed any further with tests to verify the system is a group as it is not closed.

The set of odd integers does not form a group under addition.

1

2

3

5

WORKED

E

xample

Construct a Cayley Table for [{1, i, −1, −i}, ×] and determine whether this constitutes a group.

Continued over page

THINK WRITE

Set up the empty table.

1 × 1 i 1 i

1 i −1 −i

6

WORKED

E

xample

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Note that the Cayley Table is symmetrical about the leading diagonal. The table could be flipped over on the leading diagonal and remain unchanged. This means that the order of operations will not affect the results, that is, that the operation is commutative. Therefore this group is also Abelian.

Leading diagonal

THINK WRITE

Complete the table. Remember from chapter 2 on complex numbers that

i= and i×i=−1.

Test the 4 group properties — closed set, associative, identity element and multiplicative inverse. The answers can be obtained from the table.

(Multiplication by 1 leaves all elements unchanged.)

1. All the results are members of the original set {1, i , −1, −i}. This is a closed set. 2. The set is associative

e.g. (1 ×i) ×−i=i×−i =1 and 1 × (i×−i) = 1 × 1 = 1 3. The identity element, IE×= 1

4. Multiplicative inverse: there is a 1 (IE×) in every row of the table so each element has a unique inverse.

State your conclusion. Therefore, the system is a group. 2

1 –

× 1 i −1 −i

1 i −1 −i 1 i −1 −i i −1 −i 1 −1 −i 1 ii 1 i −1 3 4

× 1 i −1 −i

1 i −1 −i 1 i −1 −i i −1 −i 1 −1 −i 1 ii 1 i −1

Construct a Cayley Table for [{mod 5}, +] and determine whether it is an Abelian group.

THINK WRITE

Decide what numbers are present in mod 5 and complete a Cayley Table of residues.

1 + 0 1 2 3 4

0 1 2 3 4 0 1 2 3 4 1 2 3 4 0 2 3 4 0 1 3 4 0 1 2 4 0 1 2 3

7

WORKED

E

xample

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175

Note: There are 9 axioms that relate to operations and whole numbers that require no proof: they are assumed to be true. The associativity statement in the example above relied on one of these axioms and you can state that these axioms have been used. They are given here with no explanation.

1. Closure Law of Addition 2. Commutative Law of Addition 3. Associative Law of Addition 4. Identity Law of Addition 5. Closure Law of Multiplication 6. Commutative Law of Multiplication 7. Associative Law of Multiplication 8. Identity Law of Multiplication

9. Distributive Law of Multiplication over addition, where a(b+c) =ab+ac

THINK WRITE

Test for the 4 group properties. 1. All results are members of the original set. So, the set is closed.

2. Addition with whole numbers is associative. 3. The identity element, IE+= 0 exists. 4. There is a 0 entry in each row because each

element has a corresponding element that, when added, results in 0 (IE+). So, there is an additive inverse.

Therefore the system forms a group.

Test for commutativity. Addition mod 5 is commutative as shown by the symmetry about the leading diagonal. For example: 4 + 0 = 4 and

0 + 4 = 4

and 4 + 2 = 1 and

2 + 4 = 1

Therefore the group is Abelian. 2

3

+ 0 1 2 3 4 0

1

2

3

4 0

1

2

3

4 1

2

3

4

0 2

3

4

0

1 3

4

0

1

2 4

0

1

2

3

remember

1. To determine whether a set forms a group under an operation (

°

) test each of the four properties; that is, test whether it is closed and associative, whether there is an identity element and a unique inverse.

2. To determine whether the group is Abelian, show that the operation is commutative (e.g. a

°

b=b

°

a).
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Properties of groups

1 a Verify that the set of real numbers, [R, +], forms a group under addition.

b Is this group Abelian?

2 a Consider the set of even numbers (2n) where n∈±Z.

b Does this form a group under addition? (Note: 0 ∉ {even numbers})

c Does it form a group under multiplication?

3 Does the set of powers of 1 form a group under:

a addition?

b multiplication?

4 Verify that the set of even integers does not form a group under division.

5 Construct a Cayley Table for [{mod 5 excluding 0}, ×] and determine whether this constitutes a group.

6 a Draw up a Cayley Table for the set of even powers of 2 under addition.

b Does this form a group under addition?

c Does this form a group under multiplication?

7 Construct a Cayley Table for [{mod 3}, ×] and determine whether it is an Abelian group.

8 Determine whether each of the tables below forms a group.

a b

c d

9 a Construct a Cayley Table for the set

a

°

b= [{5, 10, 20}, lowest common multiple of a, b]

b Does this set form a group?

10 The movements of a robot are restricted to no change (N), turn left (L), turn right (R), turn about (A): { N, L, R, A}. Construct a Cayley Table and show that this set of movements forms a group.

4C

SkillS HEET

4.1

WORKED Example 4 WORKED Example 5 WORKED Example 6 WORKED Example 7

°

a b c

a b c c a b a b c b c a

°

a b c

a b c a b c b a d c c a

°

a b c

a b c a b c b b b c c a

°

a b c

a b c b c a c a b a b c

Work SHEET

4.1

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177

Further examples

of groups —

transformations

Consider all the transformations that a shape could undergo. Rotations about its centre and reflections about its axes of symmetry involve changes in the vertices only. Carefully examine the diagram below. Make sure you understand the symbols and the new positions of the vertices. Rotations anticlockwise: R90→ 90°

R180→ 180° R270→ 270°

Reflections: RV→ in the vertical axis of symmetry RH→ in the horizontal axis of symmetry RR→ in the top right diagonal

RL→ in the top left diagonal R0→ no change.

Application of groups —

permutations

A symmetry of a square (or any other shape) may be written as a permutation by changing the positions of the vertices. For example, referring to the figure at right, we could write:

P2= , which means that vertex 1 goes to the position of vertex 2, and so on.

The only other two permutations allowed here are: P1= and P3=

Determine whether these permutations form a group under the operation

°

meaning ‘followed by’.

1

2

4 5

3 1 2 3 4 5

2 3 4 5 1

 

 

 

1 2 3 4 5 1 2 3 4 5

 

 

  1 2 3 4 5

1 5 4 3 2

 

 

 

A D

C B

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Therefore the set of all transformations or symmetries is given by the set

{R90, R180, R270, RV, RH, RR, RL, R0} and the binary operation that combines any two of these transformations is referred to as composition, where one operation follows another.

All the computerised movements involved in screen animations are based on similar compositions of transformations.

RL RL RL

R0

R 90

RV

RV RR

RR RH

RH

R 180

R 270

A B

D C

A D

B C

C D

B A

C B

D A

A D

B C

D A

C B

B A

C D

D C

A B

B

D C

A

Find the result of R180

°

Rv.

THINK WRITE/DRAW

Draw the initial square with labelled vertices.

Transform the square using R180 — 180° rotation anticlockwise. Locate vertex A and move it 180˚ anti-clockwise. All other vertices follow in order around the square.

1 D A

B C

2 B C

D A

8

WORKED

E

xample

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179

Functions

Consider functions f(x) =x, g(x) =−x, h(x) = and k(x) = (where x≠ 0).

When these functions are involved in composition of functions such as g[h(x)], the function h(x) is substituted as the inner function into the outer function which is g(x). That is g[h(x)] = − where (the inner function) is substituted into g(x) which is −(x). Similarly, k[g(x)] = − where g(x) = −x (the inner function) is substituted into

k(x) =− (the outer function). That is, k[g(x)] =− = =h(x).

THINK WRITE/DRAW

For RV mark a vertical axis of symmetry in this figure (from step 2) and reflect or ‘flip’ the square about this axis.

Reposition the vertices one side at a time, B ↔ C and A ↔ D.

This matches with a single transformation representing RH.

The result is RH.

3 B C

D A

4 C B

A D

5

1

x

--- −1

x

---1

x

--- 1

x

---1

x

( )

---1

x

--- 1

x

( )

--- 1

x

---Show that functions f(x) = x, g(x) =−x, h(x) = and k(x) = form a group under composition.

Continued over page

THINK WRITE

Complete a Cayley Table for these compositions.

1 x

--- −1

x

---1

°

f g h k

f g h k

f g h k

g f k h

h k f g

k h g f

9

WORKED

E

xample

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M a t h s Q u e s t M a t h s C Y e a r 1 1 f o r Q u e e n s l a n d

THINK WRITE

Test the 4 group properties. Closure: yes — all results are elements of the original set.

Associative: yes — for example (f

°

g)

°

h=g

°

h = k

f

°

(g

°

h) =f

°

k = k

Identity element is f(x)

Inverse: yes — f(x) occurs in every row and column.

State your conclusion. Composition of these functions forms a group. 2

3

History

of mathematics

A R T H U R C AY L E Y ( 1 8 2 1 – 1 8 9 5 )

During his life . . . Thomas Edison

invents the phonograph.

Slavery is officially abolished throughout the western world. Alfred Nobel invents dynamite.

Arthur Cayley, a famous English mathemat-ician, was born on 16 August 1821. His pub-lished mathematical papers are classics and include discussions on the concept of

n-dimensional geometry. At the age of 25 he began practising law which he continued to do until 1863. In his spare time he wrote more than 300 mathematical papers. In 1863

he accepted a professorship in mathematics at Cambridge University. One of his most famous non-mathematical accomplishments was his role in having women accepted at Cambridge.

Like Niels Abel (see page 171), many of his research topics are now used in abstract algebra and group algebra, as well as in work with matrices and the theory of determinants. The Cayley Table is named after him.

He died on 26 January 1895 having received many academic distinctions. His total works fill 13 volumes of about 600 pages each — a testimony to his prodigious life and study in mathematics.

Questions

1. What is one of Cayley’s most significant non-mathematical accomplishments? 2. List four fields of mathematics which

feature in Cayley’s work.

remember

The binary operation that combines any two transformations (for example,

rotation and reflection) is called composition, when one operation follows another.

remember

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C h a p t e r 4 A n i n t r o d u c t i o n t o g r o u p s

181

Further examples of groups

— transformations

1 a Draw a Cayley Table for the rotation of an equilateral triangle. Label each vertex.

b Does it form a group? Is it Abelian?

2 a Draw a Cayley Table for the reflections of an equilateral triangle through each of the vertices R0, RV, RL, RR.

b Does it form a group?

3 Explain what the following diagrams represent about the group shown below.

4 Describe the symmetries of the following figures, using fully annotated diagrams.

a a non-square rectangle

b a non-square rhombus

c an ellipse

5 Consider an infinitely long strip of Hs, printed on transparent paper, as shown below …..H H H H H H ….

Describe the axes of symmetry of this group.

6 Locate the axes of symmetry for the following figures.

7 a Complete a Cayley Table for the composition of the following functions.

f1(x) =x f2(x) = f3(x) =x− 1 f4(x) = where f1

°

f2=f1[f2(x)]

b Does this composition form a group?

4D

RR

RV RL

WORKED Example

8

R240

R240

F F

3

2

3 1 3

2

3

2 1

2 1

1 2 1

3

3 2

1

WORKED Example

9 1

x

--- 1

x–1

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182

M a t h s Q u e s t M a t h s C Y e a r 1 1 f o r Q u e e n s l a n d

8 a Show that the set of all 2 × 2 matrices forms a group under matrix

addition.

b i Give one example of a 2 × 2 matrix that does form a group under matrix

multiplication.

ii Give one example of a 2 × 2 matrix that does not form a group under matrix

addition.

c i Give the condition for 2 × 2 matrices to form a group under matrix

multiplication.

ii Show that these matrices form a group under matrix multiplication.

9 Show that the set of matrices forms a group

under matrix multiplication.

10 Show that the set of matrices of the form , where z is a complex number,

forms a group

a under matrix addition

b under matrix multiplication.

Assume z12 + z22≠ 0.

11 S is the set of all 2 × 2 matrices such that , where z is a non-zero complex

number.

a Show that is the identity element under matrix multiplication.

b Does the set form a group under matrix multiplication?

12 C= , where i = . The set T consists of positive powers of C such that

T =Cn where n is a positive integer.

a Find all the elements of set T.

b Does the set T form a group under matrix multiplication?

Some applications of group theory

1 Do the residues of {0, 1} mod 2 form a group under addition?

2 A teacher of abstract algebra intended to give a typist a list of 9 integers that

form a group under multiplication modulo 91. Instead, one of the 9 integers was omitted so that the list read: 1, 9, 16, 22, 53, 74, 79, 81. Which integer was left out?

3 Show that {1, 2, 3} multiplication mod 4 is not a group but {1, 2, 3, 4}

multiplication mod 5 is a group.

4 Give an example of group elements a and b with the property that a−1bab

5 The integers 5 and 15 are two of 12 integers that form a group under

multiplication mod 56. List all 12 integers.

a b

c d

1 0

0 1

1 0

0 –1

1

– 0

0 1

1

– 0

0 –1

, , ,

 

 

 

z1 z2

z2

z1

0 0

z z

0 0

1 1

i 0

0 i

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C h a p t e r 4 A n i n t r o d u c t i o n t o g r o u p s

183

6 If the following table is that of a group, fill in the blank entries.

7 Prove that if G is a group such that the square of every element is the identity, then G is Abelian.

8 Examine whether

a rotations and b reflections

as stated earlier in this section, form Abelian groups.

9 Quaternions

The concept of a set of elements called quaternions was first developed by the Irish mathematician, William Hamilton (see page 118). Quaternions are ordered sets of four ordinary numbers, satisfying special laws of equality, addition and multiplication. Quaternions are useful for studying quantities having magnitude and direction in three-dimensional space and this has enabled great advances in quantum theory, relativity, number theory and group theory.

The 4 numbers are 1, i, j and k and have the following properties: 12= 1

i2=j2=k2=ijk=−1 1i=i1

1j=j1 1k=k1

ij=−ji=k i(jk) = (ij)k=ijk

All real and complex numbers do commute with i, j, and k but they are not commutative with each other.

Follow this example that shows that jk=i ijk=−1 from the definitions

i×ijk=i×−1 multiply both sides by i on the left (or pre-multiply by i)

i2jk=−i associativity

−1 ×jk=−i from the definitions

−1 ×−1 ×jk=−1 ×−i pre-multiply both sides by −1

jk=i

Because multiplication between these elements is not commutative it is essential that all multiplication is done from a particular side of an expression and to perform this multiplication on both sides of the equal sign. You must respect the order of placement of terms in this system.

a Show that i jk=−kj ii ki=j

b Show that i–1=−i

c If q=s+wi+vj+yk and p=m+ni+oj+jk, find the product of the two quaternions.

e a b c d e

a b c

e

b c d

d

e a

e

b

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184

M a t h s Q u e s t M a t h s C Y e a r 1 1 f o r Q u e e n s l a n d

10 Pauli Matrices

The ideas introduced in the section on quaternions above can be extended to represent matrices. One 2 × 2 set is:

1 = i= j= k=

While the matrices for i and k might look a little daunting, they can be

simplified by replacing the elements with complex i.

The last three of these matrices are used in the study of quantum theories to explain and predict the behaviour of electrons. They are called the Pauli Spin Matrices and students of chemistry will appreciate the importance of the spin of electrons in atomic bonding and the strength of different materials.

A variation of these matrices used in the study of nuclear physics is shown below:

P= Q= R= S=

T= V= U= W=

On examination of the first and second rows of the matrices above you will notice that the second row is a reflection of each matrix in the first row, multiplied by i.

a Construct a Cayley Table to display the results of matrix multiplication

using these 8 matrices. Arrange them in the order given; that is, from P to W.

b Determine whether the total set forms a group.

c Mark off the top left-hand 4 × 4 corner. Examine this section of the table

and show that this subset forms a group. This is an example of a subgroup, where a subset of a group forms a complete group of its own.

11 Internet search

The real life applications of groups are quite complex. Use the internet to research this field of study. Include a list of distinct topics and a more detailed report that highlights the use of group theory.

1 0

0 1

1

– 0

0 – –1

0 1

1

– 0

0 –1

1

– 0

1 –

1 0

0 1

0 1

1

– 0

0 –1

1 0

1

– 0

0 –1

0 i

i 0

i

– 0

0 i

i 0

0 –i

0 –i

i

– 0

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C h a p t e r 4 A n i n t r o d u c t i o n t o g r o u p s

185

Groups

Modulo arithmetic is like clock arithmetic where 5 + 9 = 4 in mod 10. • The residues of modulo x are all the numbers less than x.

• Congruent numbers in mod x all differ by multiples of x.

The terminology of groups

A set S forms a group under the operation ‘

°

’ if and only if (iff) all of the following are true:

• it is closed under ‘

°

’ , that is, the result is an element of S

• the order in which operations are performed has no effect on the results, that is, it is associative

• there is only 1 identity element (IE), u, such that a

°

u=a

• there is a unique inverse a–1for every element such that a

°

a–1=u, where u= IE.

Properties of groups

• A set forms a group under an operation if elements of the set are closed and associative, and there is an identity element and a unique inverse.

• The group is an Abelian group if the operation is commutative (e.g. a

°

b=b

°

a).

Transformations

• The set of all transformations (for example, rotations and reflections) and the binary operation that combines any two of these transformations is referred to as a

composition.

summary

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M a t h s Q u e s t M a t h s C Y e a r 1 1 f o r Q u e e n s l a n d

1 Determine whether the following are groups:

a {1, −1} under multiplication b {0, 1} under addition. 2 Determine whether the following are groups:

a the set {1, 2, 4, 5, 7, 8} under multiplication modulo 9 b the set {0, 1, 2, 3, 4} under multiplication modulo 5 c the set {2, 4, 6, 8} under multiplication modulo 10 d the set {0, 1, 2} under addition modulo 3.

3 Determine whether each of the following form groups: a the set of integers where p

°

q=p+ 2q

b the set of positive rational numbers where p

°

q= .

c Show that the set of all integers forms an Abelian group under the operation

a

°

b=a+b− 3.

4 There are two lights in a room, one on the ceiling and one on the wall with 4 possible states for the two lights — both on, both off, wall light on only, or ceiling light on only. There are 4 possible changes of state: no change, both change, ceiling light change and the wall light change. These changes are denoted by N, CW, C and W respectively. Show that the set {N, C, CW, W} forms a group with respect to the operation ‘followed by’.

5 What property of a group is displayed in a Cayley Table if: a the elements are symmetrical about the leading diagonal

b the same element does not appear more than once in any row or column c the identity element occurs only once in each row or column.

6 Determine whether the following are groups: a the set of integers, modulo n under addition b the set of integers, modulo n under multiplication

c the set of integers, modulo n, excluding 0, under multiplication

d the set of integers, modulo n, excluding 0, under multiplication, if n is prime.

7 Determine whether the set of all moves that can be made by a knight on a chessboard forms a group or not.

8 a Verify that the set , where m≠ 0 forms a group under matrix multiplication.

b Verify that all p×q matrices form a group under matrix addition. 9 Show that the following set of matrices forms a group under multiplication.

10 Determine whether or not the following functions form a group under composition of functions. Assume that they are associative.

f1(x) =x f2(x) = f3(x) = 1 +x f4(x) = f5(x) = f6(x) =

CHAPTER

review

4B

4C

p q

---4C

4C

4C

4C

4D

m0 m0

4D

1 0 0 1

1

– 0

0 –1

i 0

0 i i

– 0

0 –i

0 1 1

– 0

0 –1

1 0

0 i

i 0

0 –i i

– 0

4D

1

x

--- 1

1+x

--- x

x+1

--- x+1

x

---test

test

CHAPTER

y

yourselfourself

test

yyourselfourself

4

References

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