# Symmetries and Groups

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One of the most important patterns that a mathematician looks for is whether or not an object has symmetries i.e. is left unchanged or invariant after some operation, for example reflection or rotation. A square has many symmetries under the operations of rotation and reflection whereas a rectangle has fewer symmetries. The concept of a group of symmetries measures and describes how much symmetry an object has. This concept of a group is one of the most important in mathematics and also helps to describe and explain the natural world.

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19 November 2013 Symmetries and Groups Professor Raymond Flood Welcome to the second of my lectures this academic year and thank you for coming along. This year I am taking as my theme some examples of using mathematics in various areas. One of the most important patterns that a mathematician looks for is whether or not an object has symmetries i.e. is left unchanged or invariant after some operation, for example reflection or rotation. A square has many symmetries under the operations of rotation and reflection whereas a rectangle has fewer symmetries. The concept of a group of symmetries measures and describes how much symmetry an object has. This concept of a group is one of the most important in mathematics and also helps to describe and explain the natural world. As a number such as 3 measures quantity or a number such as measures magnitude, such as the length of a line, we can think of a group as measuring symmetry. We will introduce groups by studying the symmetries of objects or structures. Mathematicians view the theory of groups as significant and beautiful. Let me give you an overview of the lecture. Group of Symmetries of the equilateral triangle The essence of a group is that it is a collection of objects with some way of combining or composing two of them together to get a third object and the way of combining or composing them has certain properties. I will illustrate this for the collection of symmetries of an equilateral triangle. In particular I will introduce the Cayley table that shows how the objects are combined. The Cayley table is named after the distinguished English mathematician, Authur Cayley. Group of symmetries of ‘six legged Isle of Man’ I will next look at the symmetries of another shape, similar to the triskelion of the Isle of Man, but with six legs to make the point I want that two groups with the same number of elements or objects can be different. Compare the group of symmetries of a square and a rectangle We will see that the group of symmetries of a square has eight elements while the group of symmetries of a rectangle has only 4 which measures how much more symmetric a square is than a rectangle. Symmetries of the platonic solids We can now with the examples we will have looked at relatively quickly show how to calculate the symmetry groups of the five regular or platonic solids – the tetrahedron, cube, octahedron, icosahedron and dodecahedron – which allows us to discover some surprising connections between the regular solids. Subgroups and Lagrange’s Theorem A subgroup is just a smaller group inside a larger one but there is major restriction on the sizes of subgroups, called Lagrange’s theorem. This has important consequences in number theory. Clock Arithmetic and Fermat’s Little Theorem Important examples of groups arise in the setting of what is sometimes called clock or modular arithmetic where the elements of the group are just the remainders you get on dividing by say 24. We will do it for a general integer and not just for the integer 24 and use Lagrange’s theorem to prove what is sometimes called Fermat’s Little Theorem. Conclusion I will end by mentioning the classification theory for a certain family of groups and a brief mention of infinite groups and some more applications. Let us start by looking at the symmetries of an equilateral triangle, this one for example although any equilateral triangle would do. We want to count all the rigid motions that leave the triangle unchanged or invariant. In a sense, I want to count the transformations with the property that if you look away and I perform one of them on the triangle then when you look back the triangle looks exactly the same. These transformations are known as symmetries. But in order to count the symmetries we do need to see what has been done and one way to do that is to label the vertices and see how they change. Here is the same equilateral triangle with the vertices labelled 1, 2 and 3. (see PowerPoint) There is a very obvious symmetry which is to do nothing – this is called the identity transformation, and is denoted by I, for identity, and although it seems quite boring it turns out to be very useful when we consider all the symmetries. Another more interesting symmetry of the equilateral triangle is to rotate it 120° anticlockwise about the centre of the triangle as shown here. Let us call this transformation R for rotation and the first thing to note is that it does leave the equilateral triangle unchanged or invariant. If we rotate by 120° anticlockwise about the centre of the triangle the triangle goes into itself with vertex 1 going to the bottom left vertex, vertex 2 going to the bottom right vertex and vertex 3 going to the top vertex. And as rotation by 120° anticlockwise is a symmetry then doing it twice i.e. rotating by 240° anticlockwise must also be a symmetry. Here you can see the effect on the vertices of rotating by 240° anticlockwise. We use the notation R2 to mean doing the rotation R twice. Doing R three times brings everything back to where it started. It is denoted by R3 and we can see from the vertices that it is just the identity symmetry, I. So R3 = I Similarly we get a symmetry if we rotate by 120° this time clockwise about the centre of the triangle or rotate by 240° clockwise. Again rotating by 360° clockwise is the identity symmetry. Another symmetric operation we can do is turning the triangle over. We call T, for turning, the transformation that turns the triangle over, keeping the top vertex fixed. This leaves the vertex at the top and the line from this vertex to the centre in the same place but swaps the other two vertices – the bottom left and bottom right as we can see on this slide. I’m sure you can suggest another couple of symmetries, for example turn the triangle over leaving the line joining the bottom left vertex to the centre fixed or turn it over leaving the line joining the bottom right vertex to the centre fixed. But I want to suggest another way of trying to find other symmetry transformations and that is to use the observation that: One symmetry transformation followed by another symmetry transformation must also be a symmetry transformation. This is because the first symmetry transformation acts on the triangle and leaves it unchanged and so does the next symmetry transformation. So doing one followed by another leaves the triangle unchanged and so is a symmetry transformation. I illustrate it on this slide by first of all doing R, rotate, and then follow it by T, turnover. We write this as TR, doing R and then T. The combination of them turns over the triangle leaving the line joining the bottom left vertex to the centre fixed. Now let us try composing these symmetries in the opposite order i.e. doing T first and then R which we write as RT. This combination of the two also turns over the triangle but this time leaving the line joining the bottom right vertex to the centre fixed. So TR is not equal to RT We say that this operation of composing symmetries doing one after the other does not commute – the order of doing things can make a difference. We now have six symmetries: I, R, R2, T, TR and RT. I: Do nothing R: rotate 120° anticlockwise about the centre of the triangle R2: rotate 240° anticlockwise about the centre of the triangle T: turns the triangle over, keeping the line joining the top vertex to the centre fixed TR: turns the triangle over, keeping the line joining the bottom left vertex to the centre fixed RT: turns the triangle over, keeping the line joining the bottom right vertex to the centre fixed. You probably remembered that I mentioned rotation clockwise about the centre of the triangle rather than anticlockwise but this symmetry is also in this list because rotating 120° clockwise is the same as rotating 240° anticlockwise. So rotating clockwise by 120 ° is just R2. But are there any other symmetries? Especially because I said that performing one symmetry followed by another gives a symmetry. So we need to check what happens when we compose each of these six symmetries with all the others and see if we get anything new. There are 36 ways to compose them. Here I have done the table of composing each symmetry with all the others. For any entry in the table we first do the symmetry for the row it is in and then the symmetry for the column it is in. You could fill in the table by drawing lots of triangles but in many cases we can use known relations. For example, the entry in red arises from doing R2 and then TR. This is TRR2 = TR3 = TI = T because R3 is I. The thing to notice is that no new symmetries appear, or as we say these symmetries of the equilateral triangle are closed under composition. This table is called the Cayley table for the symmetries of the equilateral triangle. It is named after the famous English mathematician Arthur Cayley. Here he is. From an early age, Cayley developed a remarkable ability for mathematics. At the age of 14 he enrolled as a day pupil at King’s College, London, and progressed from there to Trinity College, Cambridge. There he enjoyed a glittering undergraduate academic career, emerging top of his year and winning the coveted Smith’s prize in mathematics. With such a spectacular start to his career, he was naturally awarded a fellowship at Trinity College, but in those days college fellows were required to train for the priesthood and Cayley had no wish to do so. He left Trinity to go to Gray’s Inn, London, and train as a lawyer. During his seventeen years as a successful barrister in London, Cayley wrote over two hundred mathematical papers, including some of his most important contributions to the subject — in particular, initiating the algebra of matrices and invariant theory (the study of algebraic expressions left unchanged by certain transformations). In 1863 Cambridge University founded the Sadleirian Chair of Pure Mathematics, with no religious requirements. Cayley was duly appointed and returned to his alma mater, where he spent the rest of his life. One of the most prolific mathematicians of all time, Cayley produced almost one thousand research papers at a remarkable rate in a wide variety of topics – from algebra and geometry. Let’s look in a little more detail at the Cayley table of the symmetries of the equilateral triangle. The order of a symmetry is how many times you have to compose it with itself to get the identity. We have seen that R has order 3 because doing it three times gives the identity. The same is true for R2 – it also has order 3. Note also that every symmetry appears once and only once in every row and column – this property defines a Latin square, which is an arrangement of symbols so that so that each appears once and only once in each row and column. The symmetry T has order 2. Look at the table and combine T with T – the answer is the identity. Similarly RT and TR have order 2. This isn’t surprising as each of them turns the triangle over in a certain way and if your do it twice you get back to where you started. Now we come to one of the most important slides in the lecture – the definition of a group. What we have first of all is a set or collection of objects. Here they are symmetries of the equilateral triangle. Then we have a way of composing two of them to get a third - in this case it is doing one symmetry after another. Also there is a symmetry, called the identity, which when combined with any other symmetry leaves it unchanged. Also every symmetry has an inverse, i.e. another symmetry so that when the two are combined you get the identity. Lastly combining symmetries is associative so that if A, B and C are three symmetries then (AB)C = A(BC) In a sense this technical condition allows us to make unambiguous sense of ABC. So let us extract the essence of the definition of a group and put it more abstractly to get what is called an abstract group. We just have a set and a way of combining two members of the set to get a third and the header of the slide just gives some notation for that. But the way of combining must have three properties: First of all there is an element which does nothing. It is called the identity of the group and when combined with any other element in any order it leaves the element unchanged. Secondly if you choose any element of the group there is some element of the group so that the two of them combined in any order give the identity. Last there is a technical condition called the associative law which essentially says that we can make sense of any three elements combined together either by combining the first two first or the last two first. It gives the same answer. Well, all that was a bit abstract so let us go back to examples. Let us looks at the symmetries of this figure. It has six legs with little feet attached. It has a rotational symmetry – rotate by 60° anticlockwise about the centre, call this r. And since we know that if we do one symmetry after another we get a symmetry we then have r, r2, r3, r4, r5 and, of course, our familiar friend I, the identity which is r6. I added the little feet so that it has no reflection or turning over symmetry – because the feet would point in the opposite sense. So the symmetry group of this shape is {I, r, r2, r3, r4, r5} and here is its Cayley table. This group has six elements as had the symmetry group of the equilateral triangle. But this group and the symmetry group of the equilateral triangle are different groups! This group has 2 elements of order 6, 2 elements of order 3 and 1 element of order 2. And if you remember the group of the equilateral triangle had 2 elements of order 3 and 3 elements of order 2. Also, unlike the symmetry group of the equilateral triangle, in this symmetry group of the six legs, the order of performing the composition or multiplication does not matter. So, for example, r2 x r3 = r3 x r2 Such groups are called commutative or Abelian groups and are named after Niels Henrik Abel who was a Norwegian mathematician. Abel had rather a tragic life. He grew up in Norway and was desperate to study in the main centres of mathematical life in France and Germany, and was eventually able to obtain a stipend that enabled him to spend time in Paris and Berlin. In Germany he met Leopold Crelle and published many papers in the early issues of Crelle’s new journal, thereby helping it to become the leading German mathematical periodical of the 19th century; among these papers was the one that contained his proof of the impossibility for solving the general equation of degree 5 or more. He also obtained fundamental results on other topics - the convergence of series, elliptic functions, and ‘Abelian integrals’, many of which appeared in his ‘Paris memoir’ of 1826. The story of Abel’s attempts to be recognized by the mathematical community, and of his lack of success in securing an academic post, is a sorry one. For a time, his Paris memoir was lost. He then returned to Norway where he contracted tuberculosis and died at the early age of 26. Two days later, a letter arrived at his home, informing him that his memoir had been found and offering him a prestigious professorship in Berlin. Abel has been honoured by the International Mathematical community. A special feature of the International Congresses of Mathematics, held every four years, is the award of Fields medals to the most outstanding young mathematicians. For many years these were regarded as the mathematical equivalent of Nobel prizes, but recently a new prize, the Abel Prize, has been instituted and is awarded annually. In June 2002, to commemorate the bicentenary of Abel’s birth, the Norwegian Academy of Science and Letters launched the Abel Prize, to be presented annually by the King of Norway for outstanding scientific work in the field of mathematics. This a list of truly great modern mathematicians. 2003: Jean-Pierre Serre (France) 2004: Michael Atiyah (UK) and Isadore Singer (USA) 2005: Peter Lax (Hungary/USA) 2006: Lennart Carleson (Sweden) 2007: Srinivasa Varadhan (India/USA) 2008: John Thompson (USA) and Jacques Tits (France) 2009: Mikhail Gromov (Russia) 2010: John Tate (USA) 2011: John Milnor (USA) 2012: Endre Szemerédi (Hungary) 2013: Pierre Deligne (Belgium) Since I mentioned them in the abstract for this lecture let us quickly look at the difference in symmetries between the square and the rectangle and show how groups measure how much symmetry each has. First, the symmetries of a square. There are eight of them. On the top line we have the identity, of course, and the three rotations of 90°, 180° and 270° degrees. Below we have turning over or reflection about 4 lines of symmetry through the centre, the horizontal and vertical axes and the two diagonals. These eight symmetries form a group. If you combine any two of these eight you get another one of the eight. Also each of them has an inverse, for example rotate by 90° has inverse rotate by 270°. If you do both of them you get the identity. Now let us compare this with the symmetries of a rectangle where the adjacent sides have different lengths. Here we only have four symmetries. On the left, the identity with the little red and green circles to show the effect of the other three symmetries: rotation by 180°, and reflection in the horizontal and vertical axes. Again these four symmetries form a group – combine any two of them gives one of the four. Here I have listed out the symmetries of a rectangle and their Cayley table, calling the rotation R, the reflection or turning over in the horizontal axis, H and the reflection or turning over in the vertical axis V. If we look at the table we see that each symmetry is its own inverse – R followed by R gives the identity, H followed by H gives the identity and V followed by V gives the identity. The group is abelian or commutative, unlike the group of symmetries of a square where performing the action of rotating and then reflecting gives a different answer from reflecting first and then rotating. Let me turn to the symmetries of some solid objects the beautiful regular or Platonic solids There are only five of these polyhedral where the faces are all regular polygons and the arrangement of the faces at each corner is the same. I hope to indicate a proof that there are only these five regular polyhedral in my next lecture. The Platonic solids are: Tetrahedron – four faces each an equilateral triangle Cube – six faces, each a square Octahedron – eight faces each an equilateral triangle Dodecahedron – twelve faces each a regular pentagon Icosahedron – twenty faces each an equilateral triangle Let us count the symmetries of the tetrahedron. Choose one face as reference face then we can rotate the tetrahedron so that every other face goes to the reference face. This includes doing nothing so there are four of these. Then we can perform one of the symmetries of the equilateral triangle and we saw there are six of them. Four ways to rotate combined with six symmetries of the equilateral triangle gives 4 x 6 = 24 symmetries of the tetrahedron. We can show that this is all of them. For the cube there are six faces so six ways to rotate the cube to get to the chosen reference position and then we have seen that there are 8 symmetries of the square, the face of the cube giving in all 6 x 8 = 48. Here I have done it for all five of them and the pattern is that the number of symmetries is twice the number of faces, F, times the number of edges, E, in a face. This is because there are F ways to rotate the regular solid to a reference face and then 2E symmetries of the reference face. So, the octahedron has 48 symmetries, the dodecahedron has 120 symmetries as has the icosahedron. Note that the cube and the octahedron have the same number of symmetries. Now we know that there can be different groups with the same number of elements – we had an example of the six symmetries of the equilateral triangle and the six symmetries of the six-legged figure, but here, if we write out their Cayley tables we would see that they are the same. What do I mean by saying their Cayley tables are the same? I mean there is a way of pairing every one of the 48 members of the symmetry group of the cube with a member of the symmetry group of the octahedron and this pairing turns the Cayley table of one group into the Cayley table of the other. This pairing is called an isomorphism between the groups. So what is the connection between the cube and octahedron that makes their symmetry groups the same. It is because that inside the cube there is an octahedron and any symmetry of the cube is a symmetry of this hidden octahedron. You get this hidden octahedron by joining the centres of the faces of a cube and these form the vertices of an octahedron. But there is also a cube hidden inside an octahedron. You get this hidden cube by joining the centres of the faces of an octahedron and these form the vertices of a cube. Furthermore any symmetry of the octahedron is a symmetry of the cube. So that is why the symmetry groups of the cube and octahedron are the same. The cube and octahedron are said to be dual polyhedral. The dodecahedron and icosahedron also have the same symmetry group and a similar construction of taking the centres of each face shows that there is a dodecahedron inside an icosahedron and an icosahedron inside the dodecahedron. What about the poor tetrahedron? It turns out that the centres of its faces form another tetrahedron. It is self-dual so the construction of taking the centres of the faces gives nothing new. I want now to introduce one of the most important theorems in the theory of groups, Lagrange’s theorem. It gives us information about the size of subgroups and the size of the group in which they are found. A subgroup is a straightforward notion. A subgroup is just a little group inside the big group. It is some subset of the elements of the group that is a group itself so combining any two elements in the subset gives an element of the subset. Also the inverse of every element in the subset is also in the subset. It must also, of course, contain the identity. Here we see the Cayley table of the group of symmetries of the equilateral triangle and the subset {I, R, R2} is a subgroup of it. Let us check that: Any two elements of {I, R, R2} when combined give an element of the subset {I, R, R2}. The inverse of every element in {I, R, R2} is in the subset. They are because the inverse of R is R2 and the inverse of R2 is R. Another subgroup is {I, T} shown here. (see PowerPoint). We just need one definition. We call the order of a group the number of elements it contains. Some examples: Symmetry group of equilateral triangle has order 6 Symmetry group of square has order 8 Symmetry group of rectangle has order 4 Then Lagrange’s Theorem states that: The order of a subgroup divides the order of the group. The group of symmetries of the equilateral triangle has order 6 and the subgroup {I, R, R2} has order 3 and this divides 6. Another subgroup is {I, T} has order 2 and 2 also divides 6. The theorem is a powerful restriction on the size of subgroups. We know from it that the group of symmetries of the equilateral triangle can have no subgroup of order 4 because 4 does not divide 6. Langrange’s theorem has a powerful consequence. Recall that the order of an element was the number of times you have to combine it with itself before getting the identity. If you look at subset of all the combinations {g, g times g, g times g times g, and so on until we get the identity} it forms a subgroup with size the order of g. So by Lagrange’s theorem its order must divide the order of the group. Hence the order of any element in a group divides the order of the group. Here we see it in the case of the symmetries of the equilateral triangle where elements have order 1, 2 or 3 all dividing 6. This result has the following consequence: Consequence: Denote the order of the group as |G|. In any group any element combined with itself |G| times will give the identity. This is because the order of any element divides the order of the group so suppose the order of the group is 12, and the order of the element, g, is 4. Then g combined with itself 4 times gives the identity so g combined with itself 12 times will also give the identity. This result forms the base of public key cryptography as I will show soon. To do that I need to introduce some more groups but based around our old friends, the integers. A classic example is “clock arithmetic” where we do not consider all the integers but their remainders after dividing by 24. In general we can pick integer n. The remainders are the remainders on dividing integers by n which are {0, 1, 2… n-1} Denote the group operation by ⊕ defined as: a ⊕ b is the remainder obtained when a + b is divided by n. Suppose n is 24: The elements of the group are the members of the set {0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23} For example: 7 ⊕ 21 = 4 15 ⊕ 10 = 1 The identity is 0 as 0 added to anything gives 0. The inverse of 7 is 17 because 7 ⊕ 17 = 0. Also every element has an inverse. In fact we have seen some of these remainder groups under addition already. For the integer 3, it is isomorphic to the rotations of the equilateral triangle, for the integer 4 isomorphic to the rotations of the square and for the integer five isomorphic to the rotations of the regular pentagon. For the integer 24 it is isomorphic to the rotations of the regular 24-gon. These remainder groups under addition are important because they are the building blocks of all finite abelian groups. All finite or abelian groups can be built up from them. Next we have a group with nearly the same set of elements but a different group operation. The setup here is that we pick a prime p – a prime is an integer with no divisors other than itself and 1. An example is 7. The group operation is denoted by ⊗ and defined as a ⊗ b is the remainder obtained when a × b is divided by p. Example p = 7 Group is {1, 2, 3, 4, 5, 6} with operation ⊗ So 2 ⊗ 5 = 3 4 ⊗ 4 = 2 The identity is 1. The inverse of 2 is 4, of 3 is 5, and of 6 is 6. So it satisfies all the all the requirements or axioms of a group. I want to use this group and because it is a group we can use Lagrange’s theorem to prove something amazing. Pick your favourite prime, p. The order of the group is p-1 i.e. it has p-1elements. Pick any integer, k, smaller than p. Then k is in the group and Raise k to the power of p -1 and find its remainder on dividing by p. The answer is always 1 no matter what p or k is chosen! Pick your prime as 17 Pick your integer, k as 10. Then we know that 1017 – 1 = 1016 leaves a remainder of 1 when divided by 17. This is quite astounding. The result can easily be recast as the usual statement of Fermat’s Little Theorem (to distinguish it from Fermat’s Last Theorem): This can be recast as kp has the same remainder as k when divided by p for any prime p and integer k. My last detailed example of a group is the one that is used in the RSA system of public key cryptography. To do this I need to bring you back to your school days and ask you to remember what it means to say that two integers, such as 42 and 55, are relatively prime. It just means that they have no factors in common or that they are built from different primes. 42 = 2 x 3 x 7 while 55 = 5 x 11. Now let me show you the group. Pick an integer n. The elements of our group will be the non- zero integers less than n and relatively prime to it. The group operation is multiply together and take the remainder on dividing by n. This works because if you multiply together two numbers which have no factors in common with n then the result has no factors in common with n. If n is 24 then the group is {1, 5, 7, 11, 13, 17, 19, 23}. Identity is 1. Inverse of 5 is 5, of 7 is 7, of 11 is 11, of 13 is 13, of 17 is 17, of 19 is 19 and of 23 is 23 Every element is of order 2 in this example. On the slide e and d are two integers relatively prime to (p -1)(q – 1). You encrypt by raising the message to the power e and decrypt by raising the cryptogram to the power d. So the message is raised to the power de which gives back the message because of Lagrange’s Theorem. I have tried to give the core notions and ideas in group theory with some examples of groups and applications but of course there is much more that could be said. Let me finish with two points: First there is a certain class of groups called simple groups which can be used to build up all other groups. Simple groups are not simple things it is just that they cannot be broken down into other parts. Simple groups can be thought of very loosely as like the primes for the integers. Just as every integer is made up of primes so every finite group is made up of simple groups. One of the greatest achievements of twentieth century mathematics was the classification of all finite simple groups. When it was first done the proof took about 10,000 pages in various mathematical papers. Secondly I have only mentioned finite groups but there are also infinite groups. Think of the symmetries of a circle. You can rotate a circle through any angle about its centre and it is unchanged and there are an infinite number of angles. Or think of all the integers, positive and negative. They form an infinite group under the operation of addition. Groups and other algebraic structures are important in many areas, for example: geometry and topology, in mechanics where symmetries of the underlying equations correspond to conserved quantities, for example, the conservation of energy or the conservation of angular momentum, in the standard model for particle physics. Thank you, have a merry Christmas with lots of group activities and hope to see you in January for Surfaces and Topology. © Professor Raymond Flood 2013

## Professor Raymond Flood

### Professor of Geometry

Raymond Flood has spent most of his academic life promoting mathematics and computing to adult audiences, mainly through his position as University Lecturer at Oxford University, in the Continuing Education Department and at Kellogg College.

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