DOCSLIB.ORG

Lecture #17 of 24 ∼ November 9Th, 2020

Total Page：16

File Type：pdf, Size：1020Kb

Download full-text PDF Read full-text
Lecture #17 of 24 ∼ November 9Th, 2020 Math 5111 (Algebra 1) Lecture #17 of 24 ∼ November 9th, 2020 Products of Subgroups Products of Subgroups Semidirect Products This material represents x3.4.3-3.4.4 from the course notes. Products of Subgroups, I We proved earlier that every finitely generated abelian group decomposes as a direct product of cyclic groups. This result tells us that finitely generated abelian groups can be built up from subgroups by taking products. We can often piece other groups together from subgroups in a similar way. We would like to study how to do this, because it will give us more ways to construct finite groups and classify groups of a given order. Products of Subgroups, II If H and K are subgroups of G, then we can certainly consider the subgroup hH; Ki generated by H and K. However, the elements in this subgroup are hard to write down in general, since they are words of arbitrary length in the elements of H and K. If elements from H and K commute with one another, then by rearranging the elements in the word and using the fact that H and K are closed under multiplication, we can reduce any word to a product of the form hk for h 2 H and k 2 K. We will now look at the same set of elements for arbitrary subgroups: this is the idea of the product of two subgroups. Products of Subgroups, III Definition If H and K are subgroups of G, then the product HK is the set HK = fhk : h 2 H; k 2 Kg. The product of two subgroups is not necessarily a subgroup of G. For example, for H = f1; (1 2)g, K = f1; (1 3)g in G = S3, the product HK = f1; (1 2); (1 3); (1 3 2)g, which is not a subgroup of G. However, in some cases HK will be a subgroup: for example, with H = f1; (1 2)g and K = f1; (3 4)g in G = S4, then HK = f1; (1 2); (3 4); (1 2)(3 4)g is indeed a subgroup of G. Products of Subgroups, IV We have various properties of subgroup products: Proposition (Products of Subgroups) Let G be a group and H and K be subgroups of G. #H · #K 1. If H and K are finite, then #(HK) = . #(H \ K) 2. The product HK is a subgroup of G if and only if HK = KH. 3. If H ≤ NG (K) or K ≤ NG (H), then HK is a subgroup of G. 4. If H or K is normal in G, then HK is a subgroup of G. 5. If both H and K are normal in G, and H \ K = feg, then HK is isomorphic to the direct product H × K. 6. If np = 1 for every prime p dividing #G, then G is the (internal) direct product of its Sylow subgroups. Such groups are called nilpotent groups. Products of Subgroups, V Proofs: #H · #K 1. If H and K are finite, then #(HK) = . #(H \ K) Observe that HK is a union of left cosets of K: specifically: HK = [h2H hK. Thus we need only count how many distinct left cosets are obtained, since each left coset has cardinality #K. Consider the action of H by left multiplication on the left cosets of K in HK: by definition, there is a single orbit for this action. Products of Subgroups, VI Proofs: #H · #K 1. If H and K are finite, then #(HK) = . #(H \ K) Notice that the stabilizer of the left coset eK is the set of h 2 H with h · eK = eK, which is equivalent to saying h 2 K. Thus, the stabilizer is simply the set of h 2 H such that h 2 K, which is to say, it is the intersection H \ K. So by the orbit-stabilizer theorem, the size of the orbit is equal to the index [H : H \ K]. This means #H · #K #(HK) = #K · [H : H \ K] = , as claimed. #(H \ K) Remark: If H or K is infinite, then trivially HK is also infinite. We also emphasize that HK is not assumed to be a subgroup here. Products of Subgroups, VII Proofs: 2. The product HK is a subgroup of G if and only if HK = KH. First suppose HK = KH. Let g = hk and g 0 = h0k0 be elements of HK, with h; h0 2 H and k; k0 2 K. Then since HK = KH, the element kh0 2 KH is of the form h00k00 for some h00 2 H and k00 2 K. Then gg 0 = hkh0k0 = h(kh0)k0 = h(h00k00)k0 = (hh00)(k00k0) 2 HK. Likewise, g −1 = k−1h−1 2 KH = HK. Since the identity e = ee is clearly in HK, this means HK is a subgroup of G. Products of Subgroups, VIII Proofs: 2. The product HK is a subgroup of G if and only if HK = KH. Conversely, suppose HK is a subgroup. Then since H and K are both in HK, we have hH; Ki = HK and so KH ⊆ hH; Ki = HK. For the other containment, suppose k 2 K and h 2 H. Then we have h−1k−1 2 HK, so since HK is closed under inverses, we see (h−1k−1)−1 = kh must be in HK for any k; h. Thus, HK ⊆ KH, and so in fact HK = KH. Products of Subgroups, IX Proofs: 3. If H ≤ NG (K) or K ≤ NG (H), then HK is a subgroup of G. Suppose H ≤ NG (K), and let h 2 H and k 2 K. By hypothesis, hkh−1 2 K, and therefore we can write hk = (hkh−1)h 2 KH. Thus, hk 2 KH, and so HK ⊆ KH. Likewise, kh = h(h−1kh) 2 HK, and so KH ⊆ HK. We therefore have KH = HK, and so HK is a subgroup of G by (2). The case where K ≤ NG (H) is essentially identical. Products of Subgroups, X Proofs: 4. If H or K is normal in G, then HK is a subgroup of G. If H is normal in G, then NG (H) = G. Thus, trivially K ≤ NG (H). So by (3), HK is a subgroup of G. Likewise, if K is normal in G, then H ≤ G = NG (K), so again by (3), HK is a subgroup of G. Products of Subgroups, XI Proofs: 5. If both H and K are normal in G, and H \ K = feg, then HK is isomorphic to the direct product H × K. Since H is a normal subgroup of G, by (4) that means HK is a subgroup of G. We first show that the elements of H commute with the elements of K. To see this, observe that if h 2 H and k 2 K, then hkh−1k−1 = (hkh−1)k−1 is an element of K, since hkh−1 2 K since K is normal in G. But hkh−1k−1 = h(kh−1k−1) is also an element of H, since kh−1k−1 2 H since H is normal in G. This means hkh−1k−1 2 H \ K, and so hkh−1k−1 = e, meaning that hk = kh: thus, h and k commute. Products of Subgroups, XII Proofs: 5. If both H and K are normal in G, and H \ K = feg, then HK is isomorphic to the direct product H × K. Next, we claim that every element of HK can be written uniquely in the form hk with h 2 H and k 2 K. To see this suppose hk = h0k0 for h; h0 2 H and k; k0 2 K. Then (h0)−1h = k0k−1. But the left-hand side is an element of H while the right-hand side is an element of K, so by the assumption H \ K = feg, this common element must be the identity e. Thus (h0)−1h = e = k0k−1 and so h0 = h and k0 = k, meaning h and k are unique. Products of Subgroups, XIII Proofs: 5. If both H and K are normal in G, and H \ K = feg, then HK is isomorphic to the direct product H × K. Therefore, we have a well-defined map ' : HK ! H × K mapping hk to the ordered pair (h; k). It is a group homomorphism because if g = hk and g 0 = h0k0 then '(gg 0) = '(hkh0k0) = '(hh0kk0) = (hh0; kk0) = '(hk)'(h0k0) = '(g)'(g 0), where we used the fact that h0 and k commute. Finally, ' is trivially injective (since (h; k) = (e; e) implies hk = e) and surjective (by definition of HK) and so it is an isomorphism. Products of Subgroups, XIV A brief interjection about some terminology in this last situation. If both H and K are normal in G, and H \ K = feg, then HK is isomorphic to the direct product H × K. Under these hypotheses, we call the subgroup HK the internal direct product of H and K, and call the group H × K the external direct product of H and K. The difference is irrelevant as a practical matter, but the distinction is that the internal direct product is defined inside a group that already contains H and K as subgroups, whereas the external direct product is an explicit construction of a new group using the Cartesian product. Products of Subgroups, XV Proofs: 6. If np = 1 for every prime p dividing #G, then G is the (internal) direct product of its Sylow subgroups. The intersection of two Sylow subgroups with different primes is trivial by Lagrange's theorem, since the order of their intersection divides the order of each group.
Load more

Recommended publications
  • Representations of Certain Semidirect Product Groups* H
    JOURNAL OF FUNCTIONAL ANALYSIS 19, 339-372 (1975) Representations of Certain Semidirect Product Groups* JOSEPH A. WOLF Department of Mathematics, University of California, Berkeley, California 94720 Communicated by the Editovs Received April I, 1974 We study a class of semidirect product groups G = N . U where N is a generalized Heisenberg group and U is a generalized indefinite unitary group. This class contains the Poincare group and the parabolic subgroups of the simple Lie groups of real rank 1. The unitary representations of G and (in the unimodular cases) the Plancherel formula for G are written out. The problem of computing Mackey obstructions is completely avoided by realizing the Fock representations of N on certain U-invariant holomorphic cohomology spaces. 1. INTRODUCTION AND SUMMARY In this paper we write out the irreducible unitary representations for a type of semidirect product group that includes the Poincare group and the parabolic subgroups of simple Lie groups of real rank 1. If F is one of the four finite dimensional real division algebras, the corresponding groups in question are the Np,Q,F . {U(p, 4; F) xF+}, where FP~*: right vector space F”+Q with hermitian form h(x, y) = i xiyi - y xipi ) 1 lJ+1 N p,p,F: Im F + F”q* with (w. , ~o)(w, 4 = (w, + 7.0+ Im h(z, ,4, z. + 4, U(p, Q; F): unitary group of Fp,q, F+: subgroup of the group generated by F-scalar multiplication on Fp>*. * Research partially supported by N.S.F. Grant GP-16651. 339 Copyright 0 1975 by Academic Press, Inc.
    [Show full text]
  • Character Correspondences in Solvable Groups*
    ADVANCES IN MATHEMATICS 43, 284-306 (1982) Character Correspondences in Solvable Groups* I. M. ISAACS University of Wisconsin, Madison, Wisconsin, 53706 1. INTRODUCTION Our concern in this paper is with the ordinary (complex number) character theory of solvable groups. If the lengths of some of the earlier papers on this subject [ l-3,8] are a reliable indication, this is a surprisingly complicated and nontrivial theory. Nevertheless, some of the important results are considerably less deep than one would be led to believe by the number of pages involved. This paper contains no new theorems and few new ideas. It is addressed to the reader who, while familiar with character theory, has not mastered the literature on characters of solvable groups. Its purpose is to give such a reader a feeling for the subject and to provide him with accessible proofs for a few of the main results. Although we certainly do not attempt to prove the strongest or most general results possible (or known), nevertheless, the paper does contain a few items which have application outside of solvable groups. In particular, we have included (in Section 4) a brief exposition of tensor induction. Although our main result could have been proved using an ad hoc argument in place of this general technique, the author is now convinced (after considerable initial resistance) that tensor induction is valuable and deserves further exposure. Naturally, a key idea in the study of characters of solvable groups is to consider the relationships between characters and chief sections. In particular, suppose that K/L. is an abelian chief section of G and let ~9E Irr(K) be invariant in G.
    [Show full text]
  • Semidirect Products and Applications to Geometric Mechanics Arathoon, Philip 2019 MIMS Eprint
    Semidirect Products and Applications to Geometric Mechanics Arathoon, Philip 2019 MIMS EPrint: 2020.1 Manchester Institute for Mathematical Sciences School of Mathematics The University of Manchester Reports available from: http://eprints.maths.manchester.ac.uk/ And by contacting: The MIMS Secretary School of Mathematics The University of Manchester Manchester, M13 9PL, UK ISSN 1749-9097 SEMIDIRECT PRODUCTS AND APPLICATIONS TO GEOMETRIC MECHANICS A thesis submitted to the University of Manchester for the degree of Doctor of Philosophy in the Faculty of Science and Engineering 2019 Philip Arathoon School of Natural Sciences Department of Mathematics Contents Abstract 7 Declaration 8 Copyright 9 Acknowledgements 10 Introduction 11 1 Background Material 17 1.1 Adjoint and Coadjoint representations . 17 1.1.1 Lie algebras and trivializations . 17 1.1.2 The Adjoint representation . 18 1.1.3 The adjoint representation and some Lie theory . 21 1.1.4 The Coadjoint and coadjoint representations . 24 1.2 Symplectic reduction . 26 1.2.1 The problem setting . 26 1.2.2 Poisson reduction . 27 1.2.3 Cotangent bundle reduction of a Lie group . 29 1.2.4 Poisson manifolds and the foliation into symplectic leaves . 31 1.2.5 Hamiltonian actions and momentum maps . 33 1.2.6 Ordinary symplectic reduction . 37 1.3 Semidirect products . 41 1.3.1 Definitions and split exact sequences . 41 1.3.2 The Adjoint representation of a semidirect product . 44 1.3.3 The Coadjoint representation of a semidirect product . 46 1.3.4 The Semidirect Product Reduction by Stages theorem . 49 2 1.4 Applications to mechanics .
    [Show full text]
  • Permutation Groups and the Semidirect Product
    Permutation Groups and the Semidirect Product Dylan C. Beck Groups of Permutations of a Set Given a nonempty set X; we may consider the set SX (Fraktur \S") of bijections from X to itself. Certainly, the identity map ι : X ! X defined by ι(x) = x for every element x of X is a bijection, hence SX is nonempty. Given any two bijections σ; τ : X ! X; it follows that σ ◦ τ is a bijection from X to itself so that SX is closed under composition. Composition of functions is associative. Last, for any bijection σ : X ! X; there exists a function σ−1 : X ! X such that σ−1 ◦ σ = ι = σ ◦ σ−1: indeed, for every x in X; there exists a unique y in X such that σ(y) = x; so −1 we may define σ (x) = y: We conclude therefore that (SX ; ◦) is a (not necessarily abelian) group. We refer to SX as the symmetric group on the set X: Considering that a bijection of a set is by definition a permutation, we may sometimes call SX the group of permutations of the set X. Given that jXj < 1; there exists a bijection between X and the set f1; 2;:::; jXjg that maps an element from X uniquely to some element of f1; 2;:::; jXjg: Consequently, in order to study the group of permutations of a finite set, we may focus our attention on the permutation groups of the finite sets [n] = f1; 2; : : : ; ng for all positive integers n: We refer to the group S[n] as the symmetric group on n letters, and we adopt the shorthand Sn to denote this group.
    [Show full text]
  • Supplement. Direct Products and Semidirect Products
    Supplement: Direct Products and Semidirect Products 1 Supplement. Direct Products and Semidirect Products Note. In Section I.8 of Hungerford, we defined direct products and weak direct n products. Recall that when dealing with a finite collection of groups {Gi}i=1 then the direct product and weak direct product coincide (Hungerford, page 60). In this supplement we give results concerning recognizing when a group is a direct product of smaller groups. We also define the semidirect product and illustrate its use in classifying groups of small order. The content of this supplement is based on Sections 5.4 and 5.5 of Davis S. Dummitt and Richard M. Foote’s Abstract Algebra, 3rd Edition, John Wiley and Sons (2004). Note. Finitely generated abelian groups are classified in the Fundamental Theo- rem of Finitely Generated Abelian Groups (Theorem II.2.1). So when addressing direct products, we are mostly concerned with nonabelian groups. Notice that the following definition is “dull” if applied to an abelian group. Definition. Let G be a group, let x, y ∈ G, and let A, B be nonempty subsets of G. (1) Define [x, y]= x−1y−1xy. This is the commutator of x and y. (2) Define [A, B]= h[a,b] | a ∈ A,b ∈ Bi, the group generated by the commutators of elements from A and B where the binary operation is the same as that of group G. (3) Define G0 = {[x, y] | x, y ∈ Gi, the subgroup of G generated by the commuta- tors of elements from G under the same binary operation of G.
    [Show full text]
  • Graduate Algebra, Fall 2014 Lecture 5
    Graduate Algebra, Fall 2014 Lecture 5 Andrei Jorza 2014-09-05 1 Group Theory 1.9 Normal subgroups Example 1. Specific examples. 1. The alternating group An = ker " is a normal subgroup of Sn as " is a homomorphism. 2. For R = Q; R or C, SL(n; R) C GL(n; R). 3. Recall that for any group G, Z(G) C G and G=Z(G) is a group, which we'll identify later as the group × of inner automorphisms. If R = Z=pZ; Q; R or C then R In = Z(GL(n; R)) and denote the quotient PGL(n; R) = GL(n; R)=R× The case of SL(n; R) is more subtle as the center is the set of n-th roots of unity in R, which depends on 2 what R is. For example Z(SL(2; R)) = ±I2 but Z(SL(3; R)) = I3 while Z(GL(3; C)) = fI3; ζ3I3; ζ3 I3g. a b 0 1 a b c 0 4. But f g is not normal in GL(2;R). Indeed, if w = then w w = . 0 c 1 0 0 c b a 1 b a b 5. But f g is a normal subgroup of f g. 0 1 0 c Remark 1. If H; K ⊂ G are subgroups such that K is normal in G then HK is a subgroup of G. Indeed, KH = HK. Interlude on the big picture in the theory of finite groups We have seen that if G is a finite group and N is a normal subgroup then G=N is also a group.
    [Show full text]
  • 1 Application: Semidirect Product of (Finite) Group Schemes
    1 APPLICATION: SEMIDIRECT PRODUCT OF (FINITE) GROUP SCHEMES General assumptions and notations: • k denotes an algebraically closed field. • All k-algebras are assumed to be associative. • If A is a k-algebra, we denote by mod(A) the category of finite-dimensional mod- ules. • Given a commutative ring R, we denote by MR the category of commutative R- algebras. 1 Application: Semidirect product of (Finite) Group Schemes Assume from now on that char(k) = p > 0. A k-functor is a functor from Mk to the category of Sets. Let G be an affine group scheme over k. By definition, G : Mk −! Groups ∼ is a functor such that there is some k-algebra A and a natural equivalence G = Speck(A) of k-functors (G is representable via A). Then k[G] := A is the coordinate ring of G. The multiplication G × G −! G, the unit ek −! G and the inversion G −! G are natural transformations. Hence, Yonedas Lemma gives rise to algebra homomorphisms ∆ : A −! A ⊗k A; " : A −! k; S : A −! A. In this way, A obtains the structure of a commutative Hopf algebra (cf. [2] for some more details). 1.1 Base Change The following lemma can be proven as [1][x1, Theorem 3.5]. 0 0 Lemma 1.1. Let k ;A 2 Mk. Then the tensor product A ⊗k k of k-algebras obtains the structure of a k0-algebra via α0:a⊗β0 := a⊗α0β0. Moreover, there is a natural equivalence 0 ∼ k0 Speck0 (A ⊗k k ) = Speck(A) ◦ Resk of functors. Let now G be a group scheme with coordinate ring k[G] and fix some algebra A 2 Mk.
    [Show full text]
  • NOTES on SEMIDIRECT PRODUCTS Since Semidirect Products Are Not Discussed in the Text, Here Are Some Notes on Them. 1. Direct
    NOTES ON SEMIDIRECT PRODUCTS Since semidirect products are not discussed in the text, here are some notes on them. 1. Direct Products Let G be a group and H and K subgroups satisfying: H and K are normal in G, H \ K = f1g ;HK = G: In this case you proved in a homework and we then proved in class that G is isomorphic to the direct product H × K. Since we are happy to think of groups that are isomorphic as \essentially the same" we will simply say that G is the direct product of H and K. 2. Semidirect Products Suppose now we relax the first condition, so that H is still normal in G but K need not be. We retain the other conditions, so our conditions are now: (1) H is normal in G, (2) H \ K = f1g, (3) HK = G, In this case we obtain what is called a semidirect product. ∼ ∼ Examples: Suppose H = Z=3 and K = Z=2. There is more than one possibility for G: (i) If K is also normal in G then we already know G =∼ H × K, which is abelian and isomorphic to Z=6. (ii) On the other hand K might not be normal in G, for example G might be the symmetric group S3, with H = f(1); (123); (132)g and K = f(1); (12)g. The above examples show that for given H and K there may be more than one semidirect product. Thus, unlike for the direct product, the groups H and K may not alone give enough information to recover the structure of the group G.
    [Show full text]
  • Examples of the Atlas of Lie Groups and Representations
    Examples of the Atlas of Lie Groups and Representations Jeffrey Adams September 9, 2007 1 Introduction The Atlas of Lie Groups and Representations is a project in representation theory of real reductive groups. The main goal of the atlas computer soft- ware, currently under development, is to compute the unitary dual of any real reductive Lie group G. As a step in this direction it currently computes the admissible representations of G. The underlying mathematics of the software is described in Algorithms for Representation Theory of Real Reductive Groups [1]. This paper is a com- plement, and contains examples illustrating the algorithm and the software. 1.1 How to read this paper A brief summary of the main constructions and results of Algorithms for Representation Theory of Real Reductive Groups is given in Section 2. We recommend the reader have the paper [1] available. Unexplained notation is as in [1]. We also recommend the reader download the software from the At- las web site, www.liegroups.org; also see the www.liegroups.org/papers. The online help commands are another a useful source of information. The software is currently in an early stage of development (version 0.2.6). There will be substantial changes to the interface for version 1.0, and some additional features, which we hope to release in fall 2008. Most of the changes in the interface will be in the input methods, and we do not expect the format of ouptut to change significantly. 1 We plan to publish a manual for the software at the time version 1.0 is released.
    [Show full text]
  • Permutability of Subgroups of Semidirect Product of P-Groups 1
    International Journal of Algebra, Vol. 13, 2019, no. 7, 331 - 339 HIKARI Ltd, www.m-hikari.com https://doi.org/10.12988/ija.2019.9828 Permutability of Subgroups of Semidirect Product of p-Groups Bilal N. Al-Hasanat Department of Mathematics Al Hussein Bin Talal University Ma'an, Jordan This article is distributed under the Creative Commons by-nc-nd Attribution License. Copyright c 2019 Hikari Ltd. Abstract Two subgroups H and K of a group G are called permutable if HK = KH. Furthermore, H is permutable subgroup (quasinormal subgroup) if HK = KH for any subgroup K of G. Certainly, every normal subgroup is permutable. Permutability does not imply normality [2]. Permutability of subgroups of G × H that are direct products of subgroups of the direct factors were investigated in [6]. In this article, the classification of all permutable subgroups of the group G = Zpk oZp will be investigated for any odd prime p and for any integer k ≥ 2. On the other hand, many properties of the group G were studied and configured. Mathematics Subject Classification: 22E46, 53C35, 57S20 Keywords: Permutable subgroup, normal subgroup, group centre, gener- ator, semidirect product, capable group 1 Introduction The classification of all 2-generator p-group of nilpotency class 2 was given in [1]. The authors classified these groups using the fact that they are all central extensions of a cyclic p-group by an abelian p-group of rank 2. ∼ ∼ For groups G = Zn = hai and H = Zm = hbi, the semidirect product G o H is completely determined by how a acts on b.
    [Show full text]
  • MTHM024/MTH714U Group Theory 7 Semidirect Product
    MTHM024/MTH714U Group Theory Notes 7 Autumn 2011 7 Semidirect product 7.1 Definition and properties Let A be a normal subgroup of the group G.A complement for A in G is a subgroup H of G satisfying • HA = G; • H ∩ A = {1}. It follows that every element of G has a unique expression in the form ha for h ∈ H, a ∈ A. For, if h1a1 = h2a2, then −1 −1 h2 h1 = a2a1 ∈ H ∩ A = {1}, −1 −1 so h2 h1 = a2a1 = 1, whence h1 = h2 and a1 = a2. We are going to give a general construction for a group with a given normal sub- group and a given complement. First some properties of complements. Proposition 7.1 Let H be a complement for the normal subgroup A of G. Then (a) H =∼ G/A; (b) if G is finite then |A| · |H| = |G|. Proof (a) We have G/A = HA/A =∼ H/H ∩ A = H, the first equality because G = HA, the isomorphism by the Third Isomorphism Theo- rem, and the second equality because H ∩ A = {1}. (b) Clear. 1 Example There are two groups of order 4, namely the cyclic group C4 and the Klein group V4. Each has a normal subgroup isomorphic to C2; in the Klein group, this subgroup has a complement, but in the cyclic group it doesn’t. (The complement would be isomorphic to C2, but C4 has only one subgroup isomorphic to C2.) If A is a normal subgroup of G, then G acts on A by conjugation; the map a 7→ g−1ag is an automorphism of A.
    [Show full text]
  • Virtually Cyclic Subgroups of Three-Dimensional Crystallographic Groups
    VIRTUALLY CYCLIC SUBGROUPS OF THREE-DIMENSIONAL CRYSTALLOGRAPHIC GROUPS ANDREW GAINER, LISA LACKNEY, KATELYN PARKER, AND KIMBERLY PEARSON Abstract. An enumeration of the virtually cyclic subgroups of the three-dimensional crystallographic groups (“space groups”) is given. Additionally, we offer explanations of the underlying group theory and develop several exclusion theorems which simplify our calculations. 1. Introduction 1.1. Crystallographic Groups. The crystallographic groups may be understood geometrically, linear- algebraically, or abstract-algebraically. All three approaches will play a role in the theorems that follow, so we will outline all three here. The geometric approach is the most easily understood by intuition alone, so we will explore it first. 1.1.1. Geometric Approach. We first consider an infinite regular tiling of a Euclidean plane; similar efforts may be made on spherical or hyperbolic geometries, but this is beyond the scope of our research. We require that there be two linearly independent translations which each preserve the pattern’s symmetry— that is, that we can move the plane by specific finite distances in two distinct directions and return the pattern to its original state. Beyond these two required translations, we may be able to reflect the plane across some line or rotate it about the origin and preserve its symmetry; in some cases, it may be possible to translate and then reflect the plane such that neither the translation nor the reflection alone preserve symmetry, but their combination does— this will be termed a “glide-reflection.” In all there are seventeen possible combinations of translations, reflections, rotations, and glide-reflections which uniquely describe ways to preserve symmetries of plane patterns.
    [Show full text]