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The Cubic Groups The Cubic Groups Baccalaureate Thesis in Electrical Engineering Author: Supervisor: Sana Zunic Dr. Wolfgang Herfort 0627758 Vienna University of Technology May 13, 2010 Contents 1 Concepts from Algebra 4 1.1 Groups . 4 1.2 Subgroups . 4 1.3 Actions . 5 2 Concepts from Crystallography 6 2.1 Space Groups and their Classification . 6 2.2 Motions in R3 ............................. 8 2.3 Cubic Lattices . 9 2.4 Space Groups with a Cubic Lattice . 10 3 The Octahedral Symmetry Groups 11 3.1 The Elements of O and Oh ..................... 11 3.2 A Presentation of Oh ......................... 14 3.3 The Subgroups of Oh ......................... 14 2 Abstract After introducing basics from (mathematical) crystallography we turn to the description of the octahedral symmetry groups { the symmetry group(s) of a cube. Preface The intention of this account is to provide a description of the octahedral sym- metry groups { symmetry group(s) of the cube. We first give the basic idea (without proofs) of mathematical crystallography, namely that the 219 space groups correspond to the 7 crystal systems. After this we come to describing cubic lattices { such ones that are built from \cubic cells". Finally, among the cubic lattices, we discuss briefly the ones on which O and Oh act. After this we provide lists of the elements and the subgroups of Oh. A presentation of Oh in terms of generators and relations { using the Dynkin diagram B3 is also given. It is our hope that this account is accessible to both { the mathematician and the engineer. The picture on the title page reflects Ha¨uy'sidea of crystal structure [4]. 3 1 Concepts from Algebra 1.1 Groups Definition 1.1 A group G is an ordered pair (G; ·) where G is a set and `·' a binary operation on G satisfying the following axioms: Closure: 8a; b 2 G : a · b 2 G. Associativity: 8a; b; c 2 G :(a · b) · c = a · (b · c). Identity element: 9e 2 G 8a 2 G : a · e = e · a = a. Inverse element: 8a 2 G 9b 2 G : a · b = b · a = e. Example 1.2 Every vector space V is an abelian group with vector addition the group operation. In particular, (Rn; +), is an additive group. Example 1.3 Given a vector space V , the general linear group GL(V ) is the group of invertible linear transformations T : V ! V with composition the group operation. 1.2 Subgroups Definition 1.4 A subset H of a group G, where (H; ·) also forms a group is a subgroup of G. We denote it with H ≤ G. Example 1.5 Given a basis B of a finite dimensional real vector space V , P B L := hBiZ := f b2B z(b)b j z 2 Z g is a discrete subgroup of the additive group of V , called a lattice with basis B. Example 1.6 The orthogonal group O(V ) is defined as the set of all elements T in GL(V ) for which hT xjT yi = hxjyi is valid for any x; y 2 V . O(V ) is a compact subgroup of GL(V ). A subgroup G of O(V ) satisfies a crystallographic condition if gL ⊆ L holds 8g 2 G where L is a lattice in V . Proposition 1.7 Every subgroup G of O(V ) satisfying the crystallographic con- dition is finite. Proof: We can assume that G is a closed subgroup of O(V ) and hence is compact. The condition gL = L for g 2 G implies the existence of a matrix U(g) with integral coefficients and gB = BU(g) where B is a basis of the lattice L. Then U(g) = B−1gB holds for all g 2 G. Therefore the compact subgroup B−1GB can intersect the discrete set of all matrices with integral entries in a finite set only. Hence G is finite. Proposition 1.8 If R is a rotation in R3 that leaves a 3-dimensional lattice invariant then the angle of rotation is a multiple of either 60◦ or 90◦. 4 Proof: Similarly as in the proof of the preceding proposition we can find a regular 3 × 3-matrix B so that B−1RB is a matrix with integral entries. Now, comparing traces and observing that traceR = 2 cos φ + 1 where φ is the angle of rotation of R, we infer that 2 cos φ 2 Z. This is only possible if φ is a multiple ◦ ◦ of either 60 or 90 . Definition 1.9 When G is a group and there are subgroups H and N with N/G, G = NH and N \ H = 1 then we say that G is the semidirect product of H with N. As common, we denote the semidirect product by G = N×H. Example 1.10 For a nonsingular n × n matrix A and an element b in a fi- nite dimensional real vector space V define TA;b(x) := Ax + b. The set of all such transformations forms the affine group A(V ). There is a presentation by matrices of the form A b 0 1 Then the matrix presentation corresponding to composition TA;b ◦ TA0;b0 is ob- tained by multiplying the matrix presentations of TA;b and TA0;b0 . The trans- lations x 7! x + b constitute a normal subgroup T of G and we can present G = T ×GL(V ) as a semidirect product. Likewise the group of isometries is the semidirect product T ×O(V ) of the group of all translations x 7! x + b for b 2 V and the group of rotations. 1.3 Actions Definition 1.11 An action of a group G on a set X is a function µ : G×X ! X so that for all g; h 2 G and all x 2 X 1) µ(g; µ(h; x)) = µ(gh; x); 2) and µ(1G; x) = x. It will be convenient to write gx instead µ(g; x). A mor- phism of actions φ :(G; X) ! (H; Y ) consists of a homomorphism α : G ! H and a map β : X ! Y so that β(gx) = α(g)β(x). Example 1.12 With X := V and G := A(V ) the pair (A(V );V ) is an action. Example 1.13 The group of isometries Iso(V ) is the subgroup of all TA;b in the affine group A(V ) with hAx + bjAy + bi = hxjyi. Again, with X := V we find that (Iso(V );V ) is an action. Proposition 1.14 Let φ : V ! V be continuous and preserve length, i.e, for all x; y 2 V we have jφ(x) − φ(y)j = jx − yj. Then φ is a rigid motion. Conversely, every isometry is length preserving. Proof: Clearly, every isometry is length preserving since jOx+b−(Oy+b)j2 = jO(x − y)j2 = (x − y)T OT O(x − y) = jx − yj2. Suppose now that φ is length preserving. Then we can assume φ(0) = 0 by replacing φ by the map x 7! φ(x) − φ(0). In fact, jφ(x) − φ(0)j = jx − 0j = jxj 5 1 2 2 2 and so jφ(x)j = jxj. Since hxjyi = 2 (jx + yj − jxj − jyj ) we can deduce from this hφ(x)jφ(y)i = hxjyi to hold for all x; y 2 V . For z 2 V we have: hφ(x + y) − φ(x) − φ(y)jφ(z)i = hφ(x + y)jφ(z)i − hφ(x)jφ(z)i − hφ(y)jφ(z)i = hx + yjzi − hxjzi − hyjzi = 0: Since φ is surjective and since z 2 V can be chosen arbitrarily we conclude that φ(x + y) = φ(x) + φ(y). Then φ(mx) = mφ(x) follows by induction for all x 2 V and m 2 N. Then φ((−m)x) = φ(m(−x)) = mφ(−x) = −mφ(x) follows. 1 1 1 Replacing x by k y one can deduce φ( k y) = k φ(y). So φ(rx) = rφ(x) holds for every rational r and, by the continuity of φ it also holds for all r 2 R. Thus φ is linear and hence φ 2 O(V ). 2 Concepts from Crystallography 2.1 Space Groups and their Classification The classification of crystal symmetries amounts to classifying so-called space groups { groups of isometries of V that leave invariant a lattice in V . Definition 2.1 A discrete subgroup G of Iso(V ) is a space group provided that V=G is compact. Notation 2.2 For a space group G there is a maximal subgroup of translations x 7! x + l for l 2 L contained in it. They constitute a normal subgroup G \ TL of G. The factor group K := G=G \ TL is the point group of G. Example 2.3 Let G be generated by the symmetries of 2-dimensional space equipped with a pattern of the form ::: ∆r∆r :::. Then G \ TL would consist of all translations moving multiples of a unit ∆r. On the other hand the rotation by 180◦ around the midpoint of a unit gives rise to a subgroup of order 2 inside G. Here G can be understood as a semidirect product of its translation group and its point group. Let G be generated by a glide reflexion on 2-dimensional space with a pattern like ::: ΓLΓL ::: . Then G \ TL would consist of all translations a multiple of units ΓL. Obviously G=G \ TL is isomorphic to a group with 2 elements. Here the point group cannot be realized as a subgroup of G. Proposition 2.4 There is an induced action of the point group K on L.
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