Groups & Symmetry Robert Heffernan 1 Introduction Group theory is one of the pillars of modern mathematics and is an exciting and vibrant area of research activity. We have only a few lectures during which we will cover the absolute basics of the theory. In order to make this more exciting and to demonstrate how interesting mathematics can be we are going to aim toward a specific goal: the classification of what are called the wallpaper groups. Of course, you don’t know what a group is yet. However, we will see that while numbers aremathematicalobjectsthatmeasuresize,thereisasenseinwhichgroupsaremathematical objects that measure symmetry. A tessellation or tiling of the plane is a collection of plane figures that fills the plane with no overlaps and no gaps. Such designs can be seen in the art of M.C. Escher or in the beautiful designs of Islamic decorative art. Consider the following beautiful example of a plane-tiling from Alhambra in Spain: A symmetry of such a pattern is, loosely speaking, a way of transforming the pattern so that the pattern looks exactly the same after the transformation. We will see that different types of symmetry will give rise to different groups. These groups are called wallpaper groups or plane crystallographic groups. The aim of these notes is to prove the remarkable result that there are only 17 such wallpaper groups and, hence, there are essentially only 17 different tessellations of the plane that possess translational symmetry. This result was first proved by Evgraf Fedorov in 1891 and was independently proved by George Po´lya in 1924. It is sometimes said that all 17 of these patterns may be found in Alhambra although this may not be true. Certainly, Islamic art is a good place to look for examples of these symmetric tilings. 1 It may not be possible to finish all the material from these notes in lectures. Anything we do not cover will not be included on the examination. The slides from the lectures will be made available on WebCT. We will make use of many of the concepts we have covered up until now in the course. In particular, you should be sure that you are comfortable with sets and subsets, relations (par- ticularly equivalence relations), functions (including injections, surjections and bijections) and binary operations. You have been given a set of problems to solve. Your solutions to these will count toward your classwork grade. These problems are designed to complement these notes and spending time at them will greatly improve your understanding of the group theory we cover as well as giving you a chance to practice proving things. If you wait until the last minute to attempt these problems you will not derive very much benefit from them (both in terms of knowledge gained and marks earned). — Development 1 (M.C. Escher, 1937) 2 2 Lecture 1 2.1 Abstract groups We are already familiar with sets and with binary operations so we we will begin by giving the abstract definition of a group. We will then give some simple examples of groups. Definition 2.1. Let G be a nonempty set together with a binary operation ∗. We say that G is a group under this binary operation if the following three properties are satisfied: (Associativity) (a∗b)∗c = a∗(b∗c) for all a,b,c ∈ G; (Identity) There is an element e ∈ G such that a∗e = e∗a = a for all a ∈ G; and (Inverses) For each a ∈ G there is an element b in G such that a∗b = b∗a = e. So, to specify a group one needs two things: 1. A set G (this set can be finite or infinite); and 2. An associative binary operation ∗ on G that has an identity element e and where every element has an inverse. In our discussion of binary operations we proved the following three theorems: Theorem 2.1. A group G has exactly one identity element. Theorem 2.2. Let G be a group and let a be an element of G. Then a has a unique inverse which we denote by a−1. Theorem 2.3. In a group G the cancellation laws hold, that is • b∗a = c∗a implies b = c; and • a∗b = a∗c implies b = c for any a,b,c ∈ G. The following Lemma is also easy to prove (and useful) Lemma 2.1. Let G be a group with binary operation ∗. 1. If a ∈ G, then (a−1)−1 = a. 2. If a ,a ,...,a ∈ G, then (a ∗a ∗···a )−1 = a−1 ∗a−1 ∗···∗a−1 ∗a−1. 1 2 n 1 2 n n n−1 2 1 It is common to write ab instead of a ∗ b. We will often do this. Remember, however, that a given group G need not be commutative and so, in general, ba (cid:54)= ab. If the underlying set G is finite we call G (taken together with its binary operation) a finite group. Otherwise G is an infinite group. If a group G is finite we call |G| the order of the group G. Otherwise we say G has infinite order. 3 2.1.1 Examples (and non-examples) of groups Example 2.1. The set of integers Z is a group under the binary operation of addition. In this case the identity is 0 and the inverse of a is −a. Example 2.2. The set Q or rational numbers and the set R of real numbers are also groups under the operation of addition. Example 2.3. The set of integers Z under the binary operation of multiplication is not a group. (Why?) However, the set Q+ of positive rational numbers is a group under multiplication. In this case 1 is the identity element and the inverse if a is 1/a. Example 2.4. The set of 2 × 2 matrices of real numbers with nonzero determinant is a group under the operation of matrix multiplication. This set is usually denoted by GL (R). 2 So (cid:26)(cid:20) (cid:21)(cid:12) (cid:27) GL (R) = a b (cid:12)(cid:12)a,b,c,d ∈ R,ad−bc (cid:54)= 0 . 2 c d (cid:12) In this case the identity is the matrix (cid:20) (cid:21) 1 0 0 1 and (cid:20) a b (cid:21)−1 (cid:20) d −b (cid:21) = ad−bc ad−bc . c d −c a ad−bc ad−bc The set of all 2 × 2 matrices of real numbers is usually denoted by M (R). This is not a 2 group under matrix multiplication. Why? Example 2.5. Consider the set Z = {0,1,...,n−1} where n ≥ 1. This is a group under n the binary operation of addition modulo n. The identity element is 0 and for any j ∈ Z the n inverse of j is n−j. This group is often referred to as the group of integers modulo n and is, of course, an example of a finite group. There are many more examples of groups—the concept of a group is everywhere once you know to look for it. We will see some more examples as we go along. 2.1.2 Abelian groups Definition 2.2. A group G is called commutative or abelian if ab = ba for all a,b ∈ G. Exercise 2.1. Which of the examples in §2.1.1 are abelian? 2.1.3 Subgroups Definition 2.3. A non-empty subset H of a group G is said to be a subgroup of G if H is a group under the group operation in G. If H is a subgroup of G, then we write H ≤ G. 4 Remember that if H is a subset of the set G we write H ⊆ G. Such a subset is not necessarily a subgroup so be careful not to confuse the two notations. If H is a proper subset of the group G, ie. H ⊂ G and H is a subgroup of G, then we write H < G and call H a proper subgroup of G. We must be careful when checking to see that a subset H is a subgroup of G. Suppose G is a group with binary operation ∗. Then ∗ is a function from G × G to G. If H is a subset of G then, for H to be a subgroup, we must have that the restriction of ∗ to H ×H is a function from H ×H to H. In other words, if H is a subset of G and h ,h ∈ H then 1 2 in order for H to be a subgroup of G we must have h ∗h ∈ H. In this case we say that H 1 2 is closed under the binary operation ∗. It is easy to show that Theorem 2.4. If H is a subgroup of the group G, then the identity element of G is the same as that of H, and the inverse of each element of H is the same in G as in H. The next theorem gives a way to check if a subset H of a group G is a subgroup: Theorem 2.5. A non-empty subset H of a group G is a subgroup if and only if xy−1 ∈ H for every x,y ∈ H. Proof. It is clear that if H is a subgroup of G containing x and y, then y−1 is in H and so xy−1 is in H. Suppose conversely that H is a nonempty subset containing xy−1 whenever it contains x and y. Then, if x ∈ H we have xx−1 = e in H and so if y ∈ H, then ey−1 = y−1 ∈ H. Also, for each x,y ∈ H, x(y−1)−1 = xy is in H and so H is closed under the binary operation of G. Finally, since the associative law holds in G it also holds in H. So, we have shown that the identity element e is in H, if y is in H then y−1 is in H, H is closed under the binary operation of G and the associative law holds in H. Thus H is a group under the binary operation of G and is, as such, a subgroup of G. For any group G we have • G ≤ G; and • {e} ≤ G. 2.1.4 Cyclic groups Let G be a group. Given a ∈ G we write a1 for a and a2 for aa. Inductively, for n ∈ N, we define an+1 to be (an)a. Moreover, we define a0 to be e and for n ∈ N we define a−n to be (a−1)n. It is easy to verify that aman = am+n and (am)n = amn. Definition 2.4. A group G is called cyclic if there is an element a ∈ G such that G = {an | n ∈ Z}. Such an element a is called a generator of G. If G is a cyclic group generated by a we write G = (cid:104)a(cid:105). Example 2.6. The set of integers Z under ordinary addition is cyclic. Both 1 and −1 are generators. Example 2.7. The set Z under addition modulo n is cyclic. Again, 1 and −1 = n−1 are n generators. However, Z may have more generators than these. n 5 3 Lecture 2 (Some more abstract group theory) 3.1 Generators We can extend the notion of a cyclic group to talk about groups generated by more than one element: Definition 3.1. Let G be a group and S a subset of G. The subgroup generated by S, denoted by (cid:104)S(cid:105) is the smallest subgroup of G containing S. If S = {s ,s ,...,s } then (cid:104)S(cid:105) is the set of all products of powers of s ,s ,...,s , ie. 1 2 n 1 2 n (cid:104)S(cid:105) = {sα1sα2···sαn | α ,α ,...,α ∈ Z}. 1 2 n 1 2 n Another way to think of this is that (cid:104)S(cid:105) is the intersection of all the subgroups of G which contain S.1 Definition 3.2. If G is a group and S is a set of elements of G such that (cid:104)S(cid:105) = G, then we say that G is generated by S. So a cyclic group is a group that is generated by one element. A group may have many different generating sets.2 3.2 Homomorphisms and isomorphisms Definition 3.3. A function f from a group G into a group H is said to be a homomorphism if, for all a,b ∈ G, f(ab) = f(a)f(b). Notice that on the left-hand-side of this relation, ie. in the term f(ab), the product ab is computed in G whereas on the right-hand-side of the relation, ie. in the term f(a)f(b), the product is that of elements in H. So, a homomorphism is (loosely speaking) a function between groups that respects the binary operations of these groups. Example 3.1. If G and H are groups and e is the identity element of H, then the function H f : G → H given by f(g) = e for all x ∈ G is trivially a homomorphism. H Example 3.2. Let G be a group. The function g : G → H defined by g(x) = x for all x ∈ G is a homomorphism from G to itself. Example 3.3. Let R be the group of all real numbers under the binary operation of addition andletR∗ bethegroupofnonzerorealnumbersunderthebinaryoperationofmultiplication. Define f : R → R∗ by f(a) = 2a for each a ∈ R. To see that this is a homomorphism we must check that f(ab) = f(a)f(b), ie. that 2a+b = 2a2b which is, as we know, true. Definition 3.4. A homomorphism f from G to H is said to be an isomorphism if f is bijective. 1One of the problems in your homework is to prove that the intersection of two subgroups is a subgroup. This can easily be extended to say that the intersection of finitely many subgroups is a subgroup. 2Another homework problem asks you to show that a finite cyclic group of order n has φ(n) generators. 6 Definition 3.5. Two groups G and H are said to be isomorphic if there exists an isomor- ∼ phism between G and H. In this case we write G = H. Let’s think a minute about what it means for two groups G and H to be isomorphic. If ∼ G = H, then there exists some isomorphism f : G → H. Since f is a bijective function, the sets G and H have the same cardinality, so |G| = |H|. The isomorphism f puts each element g of G into one-to-one correspondence with some element f(g) of H and if g = g g ∈ G then f(g) = f(g g ) = f(g )f(g ). So, the sets G and H are, at least for group- 1 2 1 2 1 2 theoretical purposes, basically the same: they have the same cardinality. Moreover, the binary operations on G and H respectively are also the same. The point is this: when two groups are isomorphic, then they are, in some sense, equal. The only difference is that their elements are labelled differently. An isomorphism f gives us a way of matching the different labellings. In fact: ∼ Theorem 3.1. The binary relation = is an equivalence relation on the set of all groups: ∼ • G = G; ∼ ∼ • G = H implies H = G; and ∼ ∼ ∼ • G = H and H = J implies G = J. Proof. These follow from the (easily verified) facts that • f : G → G given by f(x) = x for all x ∈ G is an isomorphism; • If f : G → H is an isomorphism, then f−1 is an isomorphism; and • If f : G → H and g : H → J are isomorphisms, then g◦f : G → J is an isomorphism. The binary relation of isomorphism gives rise to equivalence classes which we call iso- morphism classes. From the point of view of abstract group theory two groups in the same isomorphism class are identical. However, for practical purposes we may prefer one or the other in a given situation. 3.3 Normal subgroups The following definition is of great importance in group theory (although we will not make much use of it): Definition 3.6. AsubgroupN ofagroupGissaidtobeanormal subgroupofGifwhenever g ∈ G and n ∈ N, then g−1ng ∈ N. We write N (cid:69)G. If H is a subgroup of G we often write g−1Hg to mean the set of all elements of the form g−1hg where h ∈ H, ie. g−1Hg = {g−1hg | h ∈ H}. So, a normal subgroup of G is a subgroup N such that g−1Ng ⊆ N for all g ∈ G. In fact: 7 Theorem 3.2. If N (cid:69)G, then g−1Ng = N. Proof. If g−1Ng = N for all g ∈ G then N is clearly normal. Conversely, suppose N (cid:69) G. Then if g ∈ G, g−1Ng ⊆ N and (g−1)−1Ng−1 ⊆ N, ie. gNg−1 ⊆ N. Now, since gNg−1 ⊆ N, N = g−1(gNg−1)g ⊆ g−1Ng ⊆ N and so N = g−1Ng. If you take a further course on algebra you will use the idea of a normal subgroup to define something called the quotient group. This is very important, but would take us further into group theory than we presently want to go. For now we note the following: Theorem 3.3. Let f is a group homomorphism from G to H and let e be the identity of H H. If K is the set K = {x ∈ G | f(x) = e } H then K is a normal subgroup of G. First we will prove a lemma: Lemma 3.1. If f is a homomorphism from G to H and e and e are the identity elements G H of G and H respectively, then 1. f(e ) = e ; and G H 2. f(x−1) = f(x)−1. Proof. 1. f(x)e = f(x) = f(xe ) = f(x)f(e ), so f(e ) = e . H G G G H 2. e = f(e ) = f(xx−1) = f(x)f(x−1) so f(x−1) = f(x)−1. H G Proof of Theorem 3.3. First we must check that K is a subgroup of G, then we will check that it is normal. If x,y ∈ K, then f(x) = e and f(y) = e and so f(xy) = f(x)f(y) = e e = e and H H H H H so xy ∈ K, ie. K is closed under the binary operation of G. Also, if x ∈ K then f(x) = e H and so, by Lemma 3.1, f(x−1) = f(x)−1 = e−1 = e and so x−1 ∈ K. Thus K is a subgroup H H of G. Finally suppose k ∈ K. Then f(g−1kg) = f(g−1)f(k)f(g) = f(g)−1e f(g) = e and so H H g−1kg ∈ K. Thus K (cid:69)G. The set K is called the kernel of f. It should be clear that if K = {1}, then f is an isomorphism. So, each homomorphism from G to a group H gives rise to a normal subgroup of G. It is also possible to show that each normal subgroup of G gives rise to homomorphism from G to some group H but this would again take us too far afield. 8 4 Lecture 3 (Some geometry) In this lecture we will introduce some of the geometry we need to talk about wallpaper groups (the formal definition of a wallpaper group will come later, when we’re ready for it). Since we are interested in group theory, and not geometry per se, we will state many of these results without proof.3 It might help your geometric intuition to think a little about the results below and convince yourself that they are true. I recommend that you go home and draw some pictures to convince yourself that each theorem is true (you don’t need to go so far as to write down a formal proof). We will identify the Euclidian plane with the Cartesian plane in the usual way. So, points in the plane will be represented by ordered pairs of numbers (x,y) ∈ R×R. We will denote points in the plane by capital letters P,Q,... and lines in the plane by lower-case letters l,m,.... The line segment joining a point P to a point Q will be denoted by PQ and the infinite line containing the points P and Q will be denoted by PQ. The distance from (cid:112) P = (x ,y ) to Q = (x ,y ) is defined to be d(P,Q) = (x −x )2 +(y −y )2. The origin 1 1 2 2 2 1 2 2 is the point 0 = (0,0). If a set S of points are all on some line we say that they are collinear, otherwise we say that they are non-collinear. Given points P,Q,R and S we will sometimes refer to triangles, eg. the triangle PQR: Q b P R b b and parallelograms eg. the rectangle PQRS: Q R b b b b S P A rhombus is a quadrilateral whose four sides all have the same length, eg. the rhombus PQRS: 3I will also say things like ‘it is clear that’ and ‘it is easy to show’. This is another way of saying ‘I will not bother proving...’. 9 Q R b b b b S P So, a rhombus is a special type of parallelogram and a rhombus with right angles is a square. 4.1 Transformations Definition 4.1. A transformation is is a bijective function from the set of points in the plane. If f is a transformation such that, whenever l is a line, f(l) is also a line then we call f a collineation. The identity transformation ι : R×R → R×R sends P to itself for every point P, ie. ι(P) = P for all P. Note that since transformations are bijective they have inverses and it is easy to see that the inverse of a transformation is also a transformation. Also, if α and β are transformations then the composition β ◦ α is also a transformation. Moreover, the composition of transformations is associative. So, the set S of transformations contains an identity element, is closed under composition and is closed under the taking of inverses. Thus, the set of all transformations of the plane forms a group. It is also easy to see that the set of all collineations forms a group. Wewillusuallywriteβαforβ◦αandtalkabouttheproduct (ratherthanthecomposition) of two transformations α and β. 4.2 Translations Definition 4.2. A translation is a transformation of the form (x,y) (cid:55)→ (x + a,y + b) for some fixed a,b ∈ R. Theorem 4.1. There is a unique translation taking any point P to any other point Q. Proof. If P = (c,d) and Q = (e,f) then (x,y) (cid:55)→ (x+(e−c),y +(f −d)) is a translation taking P to Q. It is easy to show that this is the only translation that will do. The unique translation taking P to Q is denoted by τ . P,Q Theorem 4.2. Each translation is a collineation. Proof. Suppose that the line l has equation ax+by+c = 0 and τ = (x,y) (cid:55)→ (x+h,y+k). P,Q Then τ = τ where R = (h,k) and the line PQ is parallel to the line 0R. So, τ (l) is P,Q 0,R P,Q the line m with equation ax+by +(c−ab−bk) = 0. Theorem 4.3. The set of all translations forms an abelian group called the translation group. 10