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 level Mandatory unit: Core Pure Major options: Mechanics Major Statistics Major Minor options: Mechanics Minor Statistics Minor Modelling with Algorithms Numerical Methods Extra Pure Further Pure with Technology Assessment Overview 144 raw marks 2 hrs 40 mins 120 raw marks 2 hrs 15 mins 60 raw marks 1 hr 15 mins (1 hr 45 mins for FPT) 50% of A level 33⅓% of A level 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 operations Two more contexts - What s the same? What s different? Mr Sticky Clock Arithmetic
Examples of binary operations Any observations?
How many groups of order 4? The two we have identified so far are: 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 How many others can you find?
Another example The operation is addition modulo 5 + 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 We write 3+ 5 4=2 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
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 Z 6 look like? What about the Cayley diagram of S 3? Explore groups further using http://groupexplorer.sourceforge.net/
Other useful sources https://plus.maths.org/content/teacher-packagegroup-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
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Modular arithmetic Z 5 What are the similarities and what are the differences between (Z 5, +) and (Z 5, )? 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 S 3. Check that the axioms for a group hold here. Is the group abelian (commutative)? Is this group isomorphic to Z n 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 (Z 8, +) 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 (Z 8, +) (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 (Z 8, +). Matrix groups 1. Show that the set of matrices of the form ( 1 n ), n N, forms an abelian group under the binary operation 0 1 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 g n? 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 z n = 1. The roots can be written as 1, ω, ω 2, ω 3,.., ω n 1 where ω = cos 2π n + i sin 2π n. By de Moivre s theorem ω k = cos 2πk n 2πk + i sin. 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 G 1 ={1, 4, 5, 6, 7, 9, 11, 16, 17} under multiplication modulo 19 and group G 2 ={0, 1, 2, 3, 4, 5, 6, 7, 8} under addition modulo 9. Composition of Functions A set consists of functions of the form f k (x) = of functions. x 1+kx for all integers k under the binary operation of composition Show that f m f n = f m+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 where the multiplications are carried out modulo 11. (x 1, y 1 ) (x 2, y 2 ) = (x 1 x 2, y 1 y 2 ) 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 and the binary operation of composition of functions. s(x) = x 1 x t(x) = x x 1 u(x) = 1 1 x Create a composition table for the group and list all of the subgroups of F. More on matrices Prove that the transformations e a b c identity reflect in the x axis reflect in the y axis 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 0 1 ), k R form a group and interpret the group geometrically. k
= {0} { a b : a, b ε Z and b 0} Q A = {xεr x 2 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 N 0 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 R 0 + N 0 \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