mei extra pure: groupsmei.org.uk › files › conference17 › session-b1.pdf · 2018-04-04 ·...
TRANSCRIPT
MEI Extra Pure:
Groups
Claire Baldwin
FMSP Central Coordinator
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