MEI Extra Pure:

Groups

Claire Baldwin

FMSP Central Coordinator

claire.baldwin@mei.org.uk

True or False activity

Sort the cards into two piles by determining

whether the statement on each card is true or

false.

MEI Extra Pure: Groups

This session will look at ways of teaching aspects

of this topic including the four simple group

axioms, Lagrange’s theorem and the concept of an

isomorphism.

The session is suitable for teachers who are

interested in learning about this Further Maths

topic or who are considering teaching MEI Extra

Pure Mathematics.

MEI Further Mathematics A levelAssessment Overview

Mandatory unit:

Core Pure

144 raw marks

2 hrs 40 mins50% of A level

Major options:

Mechanics Major

Statistics Major

120 raw marks

2 hrs 15 mins33⅓% of A level

Minor options:

Mechanics Minor

Statistics Minor

Modelling with Algorithms

Numerical Methods

Extra Pure

Further Pure with Technology

60 raw marks

1 hr 15 mins

(1 hr 45 mins for FPT)

16⅔% of A level

The content of some of the Extra Pure topics can be taught concurrently with

AS Further Mathematics

Modular MEI specification• Groups is currently on the Further Applications of

Advanced Mathematics (FP3) module

• This is a 1½ hour examination where candidates choose

3 questions from 5, worth 24 marks each.

Option 1: Vectors

Option 2: Multivariable calculus

Option 3: Differential Geometry

Option 4: Groups

Option 5: Markov Chains

• The content of the Groups section essentially the same

in the new specification with a couple of additions.

Linear MEI specification• Groups are in the Extra Pure minor option along with

Recurrence Relations, Matrices and Multivariable

calculus.

• There are no optional questions – candidates must

answer all the questions in the printed answer booklet.

• The four topics may not be evenly weighted in the

assessment e.g. on the sample assessment materials:

Q1 (10 marks) and Q2 (4 marks) – Groups

Q3 (12 marks) – Recurrence Relations

Q4 (16 marks) – Multivariable calculus

Q5 (18 marks) – Matrices

Total: 60 marks, 75 mins

Linear MEI specification

True or False activity

Sort the cards into two piles by determining

whether the statement on each card is true or

false.

True or False activity

Terminology

What do we mean by the following terms?

• a binary operation

• closed

• associative

• commutative

• identity

• inverse

Examples of binary operations

Does a Cayley /

composition table

help to analyse

the situation?

Which of the terms

on the previous

slide are relevant

here?

Examples of binary operations

x 1 -1

1 1 -1

-1 1

-1 -1 1

1 -1

The group axioms

Examples of binary operationsTwo more contexts - What’s the same? What’s different?

Clock

Arithmetic

Mr Sticky

Examples of binary operationsAny observations?

How many groups of order 4?The two we have identified so far are:

How many others can you find?

e A B C

e e A B C

A A B C e

B B C e A

C C e A B

e A B C

e e A B C

A A e C B

B B C e A

C C B A e

Another example

+ 0 1 2 3 4

0 0 1 2 3 4

1 1 2 3 4 0

2 2 3 4 0 1

3 3 4 0 1 2

4 4 0 1 2 3

Inverses: 0 is self-inverse;

1 and 4 are inverses of

each other; 2 and 3 are

inverses of each other.

NB: 1 + 4 = 2 +3 = 5

We write 3+54=2

The operation is

addition modulo 5

Activities

The activities include examples of:

• Infinite groups

• Isomorphisms

• Subgroups

• Cyclic groups

• Symmetry groups

• The order of an element

You might want to start

with the activities:

• Symmetries of an

equilateral triangle

• Permutation groups

• A group of functions

Visualising groups

Visualising groups – a Cayley

diagram

Visualising groups

Visualising groups e A B C

e e A B C

A A e C B

B B C e A

C C B A e

Visualising groups

• What would the Cayley diagram of Z6 look like?

What about the Cayley diagram of S3?

• Explore groups further using

http://groupexplorer.sourceforge.net/

Other useful sources• https://plus.maths.org/content/teacher-package-

group-theory - Teacher package of articles

introducing group theory and explaining real life

applications and the history of the subject

• Integral resources – currently available under MEI

FP3 and soon to be available on

https://2017.integralmaths.org/my/index.php (the

2017 Integral website)

• FMSP Universities page

http://furthermaths.org.uk/maths-preparation with

preparatory activities to give a taster of university

topics

Acknowledgements

• Clock Arithmetic and Mr Sticky images taken

from Maths Equals: Biographies of Women

Mathematicians, Teri Perl

About MEI

• Registered charity committed to improving

mathematics education

• Independent UK curriculum development body

• We offer continuing professional development

courses, provide specialist tuition for students

and work with employers to enhance

mathematical skills in the workplace

• We also pioneer the development of innovative

teaching and learning resources

Modular arithmetic ℤ𝟓

What are the similarities and what are the differences between (ℤ5, +) and (ℤ5,×)?

Symmetries of an equilateral triangle

A symmetry of a figure is any transformation which leaves the figure ‘looking the same’ i.e. occupying the same

area of the plane.

How many symmetries are there for an equilateral triangle?

Produce a Cayley table to show the combination / composition of the effect of two of these transformations.

The coloured dots are provided to help keep track of the orientation of the triangle.

This is called a symmetry group, usually denoted 𝑆3. Check that the axioms for a group hold here.

Is the group abelian (commutative)?

Is this group isomorphic to ℤ𝑛 under addition, where 𝑛 is the number of elements in the symmetry group (i.e. is

there a one to one correspondence between the elements of the two groups)?

Introducing subgroups

When a subset of a group forms a group in its own right, using the same binary operation, we say that the

subset is a subgroup.

Produce a Cayley table for (ℤ8, +) and check the group axioms hold. What is the identity element? What is

the inverse of each element?

What else can we say about this group?

Which of these sets are subgroups of (ℤ8, +)

(a) {0, 1, 2, 4} (b) {0, 2, 4, 6} (c) {2, 4, 6, 8}

FACT: Suppose we have a finite cyclic group of order n. For every divisor d of n, the group has exactly one

subgroup of order d.

Use this fact to identify all of the subgroups of (ℤ8, +).

Matrix groups

1. Show that the set of matrices of the form (1 𝑛0 1

) , 𝑛 ∈ ℕ, forms an abelian group under the binary operation

of matrix multiplication.

What does this group of matrices represent geometrically?

This is a cyclic group. Write down a generator 𝑔? How could we prove that any element of the group can be

written in the form 𝑔𝑛?

2. Write down the set G of matrices that represent the following transformations:

e = identity

a = rotation through 90º anticlockwise about the origin

b = rotation through 180º about the origin

c = rotation through 270º about the origin

Show that G forms a group under composition of transformations.

More symmetry groups

Construct symmetry groups for these figures:

Isosceles triangle Rectangle Square

Permutation groups

Activity adapted from Maths Equals: Biographies of Women Mathematicians, Teri Perl

The permutation dominoes below show the ways in which each of the letters in the set {A, B, C} can be paired.

Permutation dominoes can be combined by being carried out one after another. So, for example K*J would be

shown as:

The overall effect of the transformation is to map each letter to itself, i.e. K*J = I.

Show that these dominoes form a group (called a permutation group) and comment on the characteristics of

the group. To which other group is this isomorphic?

Lagrange’s theorem states that the order of a subgroup is a factor of the order of a group. Using this fact, find

the subgroups of this permutation group.

Complex roots of unity

The complex roots of unity are the solutions to the equation 𝑧𝑛 = 1.

The roots can be written as 1, 𝜔, 𝜔2, 𝜔3, … . . , 𝜔𝑛−1 where 𝜔 = cos2𝜋

𝑛+ 𝑖 sin

2𝜋

𝑛.

By de Moivre’s theorem 𝜔𝑘 = cos2𝜋𝑘

𝑛+ 𝑖 sin

2𝜋𝑘

𝑛.

Using a Cayley table show that the sixth roots of unity form a group under multiplication. To which other group

of order 6 is this group isomorphic?

Show algebraically that in the general case the nth roots of unity form a group.

Proofs on groups

Prove these results:

The identity element is unique

Each element has a unique inverse

For a group 𝐺, if 𝑎, 𝑏, 𝑐 ∈ 𝐺 and 𝑎𝑏 = 𝑎𝑐 then 𝑏 = 𝑐 [this is called the cancellation property for groups]

Specifying an isomorphism

Specify two distinct isomorphisms between the group 𝐺1={1, 4, 5, 6, 7, 9, 11, 16, 17} under multiplication

modulo 19 and group 𝐺2={0, 1, 2, 3, 4, 5, 6, 7, 8} under addition modulo 9.

Composition of Functions

A set consists of functions of the form 𝑓𝑘(𝑥) =𝑥

1+𝑘𝑥 for all integers 𝑘 under the binary operation of composition

of functions.

Show that 𝑓𝑚𝑓𝑛 = 𝑓𝑚+𝑛 and hence show that the binary operation is associative.

Show that this set of functions forms a group

State one subgroup of this group (other than the trivial subgroup and the whole group)

Cyclic subgroups

The set G = {1, 3, 4, 5, 9} forms a group under multiplication modulo 11.

The set H consists of the ordered pairs (𝑥, 𝑦) where 𝑥, 𝑦 are elements of the group G and the binary operation

is defined by

(𝑥1, 𝑦1) ∗ (𝑥2, 𝑦2) = (𝑥1𝑥2, 𝑦1𝑦2)

where the multiplications are carried out modulo 11.

What is the identity element of H?

Is it true that (𝑥, 𝑦)5 = (1,1) for each element in H?

Suppose a subgroup of H has order 5 and contains the element (4,5) – list the other elements of this

subgroup

How many subgroups of H would there be with order 5?

A group of functions

The group F = {p, q, r, s, t, u} consists of the six functions defined by

p(𝑥) = 𝑥 q(𝑥) = 1 − 𝑥 r(𝑥) = 1

𝑥 s(𝑥) =

𝑥−1

𝑥 t(𝑥) =

𝑥

𝑥−1 u(𝑥) =

1

1−𝑥

and the binary operation of composition of functions.

Create a composition table for the group and list all of the subgroups of F.

More on matrices

Prove that the transformations

𝑒 identity

a reflect in the 𝑥 axis

b reflect in the 𝑦 axis

c rotate through 180° about the origin

form a group and write down a corresponding matrix group to represent the same transformations.

Prove that the set of matrices of the form (𝑘 0

01

𝑘

) , 𝑘 ∈ ℝ form a group and interpret the group geometrically.

∅ = {0} {𝑎

𝑏: 𝑎, 𝑏 𝜖 ℤ and 𝑏 ≠ 0} ⊆ ℚ

𝐴 = {𝑥𝜖ℝ ∶ 𝑥2 ≥ 90}

9.5 ∈ 𝐴

0 ∉ ℕ 𝑦 ∈ {𝑦} 𝑥 ∈ {𝑦}

𝐴 = {𝑥: 𝑥 is even and 𝑥 < 20} 𝐵 = {𝑥: 𝑥 is prime and 𝑥 < 30}

𝐴 ∩ 𝐵 = ∅

0 ∉ ℕ0

𝐴 = {𝑥: 𝑥 is prime and 𝑥 < 50}

𝑛(𝐴) = 15

𝐴 = {𝑥: 𝑥 is even and 𝑥 < 20} 𝐵 = {𝑥: 𝑥 is prime and 𝑥 < 30}

𝐴 ⊄ 𝐵

𝐴 = {𝑥: 𝑥 is even and 𝑥 < 20} 𝐵 = {𝑥: 𝑥 is prime and 𝑥 < 30}

𝑛(𝐴 ∪ 𝐵) = 17

−3 𝜖 ℝ0+

ℕ0\ℕ = {0} 𝐴 = {𝑇he letters in the word NULL}

𝐵 = {The letters in the word FINITE}

𝐴 ∩ 𝐵 = ∅

𝐴 = {𝑥: 𝑥 𝜖 ℕ, 𝑥 is prime} 𝐴 is a finite set