Decomposition of Sohncke Space Groups Into Products of Bieberbach and Symmorphic Parts

Z. Kristallogr. 2015; 230(12): 719–741

Gregory S. Chirikjian*, Kushan Ratnayake and Sajdeh Sajjadi Decomposition of Sohncke space groups into products of Bieberbach and symmorphic parts

DOI 10.1515/zkri-2015-1859 only the identity and torsion elements (roto-reflections).1 Received May 21, 2015; accepted October 16, 2015 In the symmorphic case, the Bieberbach subgroup ΓB with smallest index in Γ is simply T, the translation group of Abstract: Point groups consist of rotations, reflections, the primitive lattice. And in this case it is well known that and roto-reflections and are foundational in crystallog- Γ = T  P (the semi-direct product of T and P) where P is raphy. Symmorphic space groups are those that can be P the point group. We have observed that for many nonsym- decomposed as a semi-direct product of pure translations morphic subgroups there is a Bieberbach group Γ such and pure point subgroups. In contrast, Bieberbach groups B that [Γ:Γ ] < [Γ:T] (where [G:H] denotes the index of H in consist of pure translations, screws, and glides. These B G) and “torsion-free” space groups are rarely mentioned as being ΓΓ=< S where SP. (1) a special class outside of the mathematics literature. Every B space group can be thought of as lying along a spectrum (Here < denotes “is a proper subgroup of”.) In particular, with the symmorphic case at one extreme and Bieberbach in this paper we focus on the 65 Sohncke space groups space groups at the other. The remaining nonsymmorphic that preserve the chirality of crystallographic motifs. space groups lie somewhere in between. Many of these In our calculations we work in the factor group Γ/Σ can be decomposed into semi-direct products of Bieber- where Σ ≤ T is the translation group of a sublattice of the bach subgroups and point transformations. In particu- primitive lattice that is normal in Γ, and we arrive at state- lar, we show that those 3D Sohncke space groups most ments such as2 populated by macromolecular crystals obey such decom- Γ Γ ==BS where B: B (2) positions. We tabulate these decompositions for those ΣΣ Sohncke groups that admit such decompositions. This (which we call a semi-decomposition), and we show that has implications to the study of packing arrangements the equalities in (1) and (2) are equivalent. The fraction in macromolecular crystals. We also observe that every notation is used to emphasizes that Γ/Σ = Σ\Γ, which Sohncke group can be written as a product of Bieberbach holds when Σ is a normal subgroup. (Normality of a sub- and symmorphic subgroups, and this has implications for group is denoted as Σ  Γ.) new nomenclature for space groups. The primary goal of this paper is to identify which Keywords: Bieberbach groups; crystallographic space Sohncke groups can be decomposed as in (1) and to iden- groups; point groups. tify ΓB and S in these cases. We have found that all but four Sohncke groups can be decomposed in this way. As a secondary goal for these four we seek a weaker decompo- Introduction sition of the form

ΓΓ= BS (3) In any space group Γ there exist Bieberbach (torsion-free) where Γ need not be normal. We find that two of these four subgroups (consisting only of translations, screw dis- B can be decomposed in this way, and that (3) is backwards placements, and glides) and subgroups which contain 1 In general, an element g of a discrete group G is called a torsion element if gn = e (the group identity) for some finite natural number n. *Corresponding author: Gregory S. Chirikjian, Department of That is, a torsion element is an element of finite order. For any space Mechanical Engineering, The Johns Hopkins University, 3400 N. group together with an appropriate choice of origin, each torsion Charles Street, Baltimore, MD 21218, USA, element generates a cyclic subgroup of S: = {0}  S < Γ where S is a E-mail: [email protected] subgroup of the point group, P. Kushan Ratnayake: The Johns Hopkins University, 3400 N. Charles 2 In fact, most of the time it is possible to take Σ = T, but there are a Street, Baltimore, MD 21218, USA few cases where taking Σ < T enables the decomposition to be per- Sajdeh Sajjadi: Department of Mechanical Engineering, The Johns formed when it would not be possible otherwise, as will be described

Hopkins University, 3400 N. Charles Street, Baltimore, MD 21218, USA later in the paper. (An example of this is P42.)

Brought to you by | De Gruyter / TCS Authenticated Download Date | 12/3/15 7:49 PM 720 G.S. Chirikjian et al.: Decomposition of Sohncke space groups into Bieberbach and symmorphic parts compatible with (1). That is, whenever (1) holds, so must atoms are introduced. The spirit of molecular replacement (3), but not the other way around. is that if the model is a good shape match for the actual We also observe that every Sohncke space group can protein molecule of interest, and if it has similar density, be written as a product of the form then the phase problem can be solved computationally rather than experimentally. Given such a model molecule, ΓΓ= Γ (4) B S it should be possible to find a translation and rotation,3 3  where ΓB and ΓS are respectively Bieberbach and sym- g = (R, t) ∈ SE(3): = ℝ SO(3) (the group consisting of all morphic subgroups, regardless of whether or not there special Euclidean motions), to place the model molecule is a normal ΓB, and (4) need not correspond to the above in the unit cell in the same way as the actual molecule. decompositions, though it is backwards compatible with Here ℝ3 is three-dimensional Euclidean space, SO(3) is them. That is, whenever (3) holds it must also be true that the group of rotations (i.e. 3 × 3 orthogonal matrices with (4) holds, but not vice versa. positive determinant, thereby excluding reflections), and The remainder of this paper is structured as follows. The SE(3) is the full group of rigid-body motions (which is a section “Motivation: the molecular replacement problem” six-dimensional Lie group). Of course, the pose sought in describes our motivation in pursuing these decomposi- MR is not unique because both the symmetry mates within tions. The section “Literature review” reviews the literature a unit cell and translates in all of the other unit cells are to explain where the current work fits in the broader world equivalent to this pose. This equivalence is described by of mathematical crystallography. In the section “Definitions them being members of the same right coset Γg where and core theorems” a small subset of the fundamental defi- Γ < SE(3) is the Sohncke space group of the crystal. The nitions and concepts from group theory most relevant to our group element g ∈ SE(3) appears on the right because it study are reviewed. The section “Theory behind the decom- is rigid motion relative to the motions described by the posability of space groups” presents a new theorem to assist globally defined motions describing the crystal symmetry. in our calculations. The section “Outline of decomposition Therefore, one does not need to search the full space SE(3) algorithms for Sohncke groups” outlines the algorithmic in MR. Rather, choosing one representative element of approach that we take to performing these decompositions. SE(3) from each of the right cosets in the right coset space In the section “Table of semi-decomposable nonsymmor- Γ\SE(3) will describe all nonredundant ways to situate a phic Sohncke space groups” tables describing the complete model molecule. We collect all such g’s into a connected results of these calculations are given. and bounded six-dimensional subset of SE(3) denoted as ℱ Γ\SE(3). (In general such a set of coset representatives is called a fundamental domain.) We then take the closure of this set, which adds some redundancy in the same Motivation: the molecular way that choosing a closed interval with boundary points replacement problem identified is slightly redundant as a fundamental domain for a the circle group ℝ/(2πℤ). We denote this closure of ℱ̅ In a series of papers [1–3] we are developing new methods the fundamental domain as Γ\SE(3) ⊂ SE(3). ℱ̅ for molecular replacement (MR) [4] for use in the field Explicit choices for Γ\SE(3) are described in [2]. For of macromolecular crystallography. This motivated the example if kind of decomposition proposed in this paper for certain ΓP :,=T P (5) of the Sohncke groups (i.e. the 65 of the 230 types that then we can choose preserve chirality). Though the methodology presented here is applicable to many non-Sohncke groups as well, FF=×3 F ΓP\(SE 3) T\R P\(SO 3) we limit the current presentation for issues of brevity and [where here × is the Cartesian product, rather than direct relevance to the MR application. product, and ℱ ̅ ⊂ SO(3)]. This makes sense because In MR, a diffraction pattern (or equivalently, the P\SO(3) P < SO(3). Alternatively we can always make the choice for Patterson function) for a macromolecular crystal is pre- any Sohncke space group Γ sented, and a model for the hypothesized shape of the macromolecule (which is typically a protein) is con- FF=×3 SO(3). Γ\(SE 3) Γ\R structed from information in the Protein Data Bank Since the choice of fundamental domain can be made in a [5]. The problem is that the diffraction pattern does not number of ways based on properties of Γ, this motivates us contain phase information. Such information is often provided by additional experiments in which heavy metal 3 The combination of which is called “pose” for brevity.

Brought to you by | De Gruyter / TCS Authenticated Download Date | 12/3/15 7:49 PM G.S. Chirikjian et al.: Decomposition of Sohncke space groups into Bieberbach and symmorphic parts 721

 to investigate the structure of Γ. For example, if Γ = ΓB S whether Γ is Bieberbach or not, the fundamental domain 3 where < , then we can make the additional choice F 3 ⊂ can be identified with the crystallographic S P Γ\R R

asymmetric unit. When faces of F 3 are glued appropri- Γ\R FF=×3 F . Γ\(SE 3) Γ \ S\(SO 3) 3 B R ately, the result is Γ\ℝ . For this reason, the modern geometric approaches to crystallography have relevance to This is potentially interesting because the space Γ \ℝ3 is B structural biology. Moreover, if E(3) denotes the noncom- an “orientable flat manifold” and the coset space S\SO(3) pact noncommutative six-dimensional Lie group of all can be identified with a “spherical space form,” which is Euclidean motions and Γ is any space group drawn from also orientable. the 230 types, then each Γ\E(3) is a compact six-dimen- Though the non-Sohncke case is not relevant to sional manifold. These manifolds (and their generaliza- macromolecular crystals, one can ask analogous ques- tions in higher dimensions) were mentioned by Hilbert in tions about how the fundamental domains corresponding the formulation of his “18th problem,” but other than the to Γ\E(3) can be defined where fact that they are known to be compact, they appear not E(3):= R3 O(3) to have been studied since. In the Sohncke case, Γ\SE(3) is the configuration space of a rigid-protein molecule is the full Euclidean group (as opposed to the “Special” moving in Γ\ℝ3. The decompositions presented in this Euclidean group), including reflections and glides. But in paper allow us to better understand the structure of this the current paper we focus only on the case of Sohncke configuration space to address the molecular replacement groups. problem with new computational tools. In our work, we use two theorems given later in the paper in combination with functions that we use in the Bilbao Crystallographic Server [45–47] (particularly the Literature review COSET, MAXSUB, SUBGROUPGRAPH, IDENTIFY GROUP, and HERMANN functions). We have made extensive use In recent years there appears to be a renaissance in the of this valuable community tool to both refine our theory field of mathematical crystallography, as described in and to do computations. [6–8]. Mathematical and computational crystallography has many facets including algebraic structure theory of space groups [9–13], geometric approaches [13–24] and topologi- Definitions and core theorems cal approaches [12, 25]. And the representation theory of space groups has long been known to play an important We begin with a few basic definitions in order to establish role in solid state physics and therefore has been studied notation, under the assumption that the reader is already intensively [26–29]. familiar with the basic concepts of group theory. Many excellent general introductions to mathematical crystallography are available including [29–35] as well as introductions to group theory [36, 37] and books on solid- The most basic definitions state physics and phase transitions [38–40] that have detailed mathematical introductions to crystallography. Let G denote a group. If G is finite, then |G| denotes the Of course, the most complete source for knowledge about number of elements in it. The product of two elements g, mathematical properties of space groups is the Interna- h ∈ G is denoted by simply writing the elements next to tional Tables, and particularly Vols A and A1 [41, 42]. each other, gh. In general gh ≠ hg. The identity element is The space-group decompositions derived here origi- denoted as e ∈ G, and the inverse of an element g ∈ G is nated in our study of the molecular replacement problem denoted as g−1 ∈ G. in protein crystallography, as described in [1–3]. In partic- Given a group G, two elements a and b are called con- ular, in that problem the quotient space in which copies 3 jugate if there exists an element g such that of proteins are packed is Γ\ℝ . When Γ = ΓB (a Bieberbach group), it acts properly discontinuously and freely on ℝ3 agg :.==ag −1 b (6) 3 and so ΓB\ℝ is a flat manifold [14, 24, 43] which is orientable if and only if ΓB is Sohncke . More generally, the The notation H < G indicates that H is a proper sub- space Γ\ℝ3 is a Euclidean orbifold [22, 44]. Regardless of group, i.e. a subgroup for which H ≠ G. In contrast, the

Brought to you by | De Gruyter / TCS Authenticated Download Date | 12/3/15 7:49 PM 722 G.S. Chirikjian et al.: Decomposition of Sohncke space groups into Bieberbach and symmorphic parts

notation H ≤ G indicates that H is a subgroup, leaving and H is a space subgroup, the fundamental domain FG/H open the possibility that H = G. will be a finite set. Two subgroups H, K < G are called conjugate if there exists an element g ∈ G such that Space groups H g :.==gHgK−1 (7) In our study, we are interested in space groups, denoted If in the above expression Hg = H for all g ∈ G, then H < G is as Γ, acting on three-dimensional Euclidean space. These called a normal subgroup of G and we write H  G. discrete groups are subgroups of E(3), the group consist- Given a subgroup H < G, and any element g ∈ G, the ing of the continuum of all possible rigid-body motions of left coset gH is defined as three-dimensional Euclidean space. In turn, E(3) < Aff(3), gH:{=∈gh|}hH. the group of affine transformations. All three groups Γ < E(3) < Aff(3) can be thought of as consisting of ele- Similarly, the right coset Hg is defined as ments that are pairs of the form (A, a) where A ∈ GL(3, Hgh:{=∈gh|}H . ℝ) (the general linear group consisting of 3 × 3 invertible matrices with real entries) and a ∈ ℝ3. And the group oper- A group is divided into cosets (left or right) all of the same ation is5 size. And the set of all cosets is called a coset space. The left coset space (,AA11 aa)( 22, )(=+AA12, A12aa1 ). (8) GH/:=∈{|gH gG} This is equivalent to expressing group elements as 4 × 4 homogeneous transformation matrices of the form and the right coset space A a H\:GH=∈{|gg G} H(,A a) = t 0 1 are generally different from each other. But when H  G, then gH = Hg for all g ∈ G and G/H = H\G. In this special (where 0t is the transpose of the 3-dimensional zero G case we can denote both as , and this space forms a vector), and describing the group operation as matrix H multiplication. group under the operation (g H)(g H) = (g g )H. 1 2 1 2 The relationship between (A, a) and H(A, a) is an Given a left coset decomposition, it is possible to example of an isomorphism. That is, there is a bijective define (in a non-unique way) a fundamental domain correspondence between the set of all pairs of the form (A, F H GH/ ⊂ G a) and of all matrices of the form (A, a), and the group operation is preserved by this correspondence in the consisting of exactly one element per left coset:4 sense that F HH= H ||GH/ ∩=gH 1 (( AA11, aa)( 22, )) (,AA11 aa)(22, ). for every g ∈ G. (And similarly for right-coset decomposi- When two groups G and G′ are related by an isomorphism tions.) Since a group is partitioned into disjoint cosets, they are called isomorphic, and this is written symboli- cally as G ≅ G′. ∪ gH = G. F Throughout this paper we describe three-dimensional g∈ GH/ crystallographic group operations by their actions on an In the case when H  G, this fundamendamental domain arbitrary point x ∈ ℝ3. That is, instead of writing γ ∈ Γ is a group with respect to the original group operation explicitly as a pair of the form (A, a) or as a matrix H(A, a) mod H. And this group is isomorphic with G/H. The utility we will often write the row vector (γ · x)t. This is consistent of this concept is that we can use it to perform set-theo- with the way space group operations are reported in the retic calculations inside of G. crystallography literature. Explicitly, When G is a Lie group such as SE(3) and H is a discrete γ =+(,RRγγ tv()γ ) subgroup, the fundamental domain FG/H will have the same dimensionality as G. But when G is a space group 5 Various other notations for (A, a) include (A|a), {A|a} and versions 4 For any finite set, X, the number of elements in X is denoted as |X|. of these with A and a written in reverse order.

Brought to you by | De Gruyter / TCS Authenticated Download Date | 12/3/15 7:49 PM G.S. Chirikjian et al.: Decomposition of Sohncke space groups into Bieberbach and symmorphic parts 723

F and Γ/Tp:{=∈(,RR v()pp)|R P} (12)

γ⋅=xxRRγγ++tv()γ . where v(Rp) is a translation by a fraction of the unit-cell dimensions corresponding to a screw displacement or

Here Rγ ∈ P (the point group), tγ ∈ L (the lattice) and glide. The ≅ notation hides v(Rp) from view, and therefore v:P → ℝn is a translation by a fraction of a lattice motion should not be misinterpreted as equality. ℱ such as those that can appear in screw or glide motions, When v(Rp) ≠ 0 for at least one Rp ∈ P, then Γ/T is not and the mapping v observes the co-cycle condition a group under the original operation of Γ. However, it does become a group under the rule vv()RR =+RR()v()RT mod . (9) γγ12 γγ12 γ1 (,RR vv( ))( RR, ()): = (,RR v()RR ) pp11pp22 pp12 pp12 The “mod T” effectively removes any component in the sum that is in T, in analogy with the way 3 + 4 = 2 mod 5 in 3 where (9) is used to evaluate v()RRpp. modulo 5 arithmetic. If there exists a point p ∈ ℝ such that 12 But since the group operation of ℱ is not that of Γ, it γ · p = p for some γ ∈ Γ other than the identity element, Γ/T follows that ℱ cannot be a subgroup of Γ in the nonsym- then γ is called a torsion element of Γ. If it is possible to Γ/T morphic case. choose a coordinate system in ℝ3 with p as the origin such With the above defined group operation, the follow- that v(R ) = 0 for all γ ∈ Γ, then Γ can be decomposed as γ ing isomorphism holds a product of translations and torsion elements (rotations, Γ reflections, and roto-translations). That is, Γ = T · P where F Γ/T ≅ (13) all of the elements of T P:{= 0} P (10) and this is somewhat more descriptive of Γ than is (11) ℱ preserve the location of the origin. The distinction between because the set Γ/T can be used to reconstruct Γ as P and P is that P < Γ whereas P is not. Γ = T · P is the sym- Γ =⋅T F (14) morphic case in which the group is decomposed into a Γ/T product of translations and point-preserving transforma- where the product in the above expression simply means tions, P. Of the 230 types of 3D space groups, 73 can be that every γ ∈ Γ can be decomposed as decomposed in this way. These are the symmorphic space (,RR tv+=())(I, tv)( RR, ()). groups. Some space groups have no torsion elements. γγ γγγγ These are the Bieberbach groups. For them it is not pos- Note that even in the symmorphic case where v(R ) = 0 sible to define an origin such that v(R ) = 0 for any γ ∈ Γ p γ for all R ∈ P, (13)–(14) and (11) are not the same in the other than the identity element. p F strictest possible sense. But at least in this case Γ /T : = P In the theory of space groups, perhaps the most well- P is a subgroup of Γ . Here we have used the definitions in known isomorphism originates from the fact that the P (5) and (10). translation group of the primitive lattice is always normal, To illustrate further the difference between isomor- T  Γ, and phism and equality, consider the following. If H  G and Γ K < G such that H ∩ K = {e}, and HK = G then G = H  K. ≅ P. (11) T Corresponding to the coset space G/H = H\G is the fundamental domain constructed from one representative F element of G per coset: G ⊂ G. There is no unique way to H Isomorphism vs. equality of fundamental construct this fundamental domain, but a natural choice domains is F G :,= K Though the concept of isomorphism is critically impor- H tant in group theory, we stress the difference between G which is consistent with the isomorphism ≅ K, but isomorphism and true equality by considering fundamen- H ℱ tal domains of the form Γ/T constructed by choosing one is stronger than isomorphism in the sense that it is an element per coset in Γ/T. In the case when coset repre- equality. sentatives are chosen with minimal translational part, a If in addition there is a supergroup A > G, then for any stronger version of (11) can be written by defining α ∈ A we can write

Brought to you by | De Gruyter / TCS Authenticated Download Date | 12/3/15 7:49 PM 724 G.S. Chirikjian et al.: Decomposition of Sohncke space groups into Bieberbach and symmorphic parts

αGHαα−−11= ()αα()Kα−1 But either way, ααGG−1 in which case we can choose ≅ . ααH −1 H −1 F −1 :.= ααK ααG ααH −1 Classification of space groups In the context of symmorphic space groups G: = ΓP , if H: = T and K: = {0}  P and if A = Aff(3), then it is possible A space group is called Sohncke if det R = +1 for all γ ∈ Γ. to find α ∈ A such that Σ = αTα−1 is the translation group γ In the table below we classify space group types. Since of a sublattice. Moreover, it can be the case that αKα−1 = K the translation group T is both Bieberbach and sym- when α ≠ e. For example, if morphic (with trivial point group) it can be classified as 1000 both symmorphic and Bieberbach. We include T with the  0100 symmorphic space groups and call the remainder of the α =  00a 0 Bieberbach groups “nontrivial.”  000 1 and Symmorphic Nontrivial NonSymmorphic + Total Bieberbach nonBieberbach  csθθ−+0 vx11wy Sohncke 24 8 (5) 33 (25) 65 (54) scθθ0 vx+ wy Non- 49 4 112 165 γ = 22 001 z Sohncke Total 73 12 (9) 145 (137) 230 (219) 0001 where θ takes discrete values consistent with the 2D The number in parenthesis is for types classified t t lattice spanned by v = [v1, v2] and w = [w1, w2] and x, y, according to equivanence under conjugation by general z ∈ ℤ, and a ∈ ℤ > 0, then affine (rather than proper affine) transformations. That is, when counting the space group types in parenthesis,  csθθ−+0 vx11wy the following enantiomorphic pairs are each considered scθθ0 vx+ wy −1 22 αγα = . as being of the same type: (P31, P32), (P41, P43), (P61, P65), 001 az (P6 , P6 ), (P3 12, P3 12), (P3 21, P3 21), (P4 22, P4 22), 2 4 1 2 1 2 1 3 0001 (P41212, P43212), (P4132, P4332), (P6122, P6522), (P6222, P6422). Note that this distinction is only important in the case of If P denotes the point group and P denotes the subgroup Sohncke space groups. of Γ defined in (10) consisting of all transformations of the form of γ given above when x = y = z = 0, then for the present example it is clear that

−1 Theory behind the decomposability αPPα = . Therefore, similar to how for any natural number, n, of space groups nℤ can be both a subgroup of ℤ and isomorphic with it, The main goal of this paper is to determine which space both of the following statements can be true simultane- groups can be decomposed as ously: αGα−1 ≅ G and αGα−1 < G (which is an example illus- trating how isomorphism is weaker than equality). In the ΓΓ=

FF1 ==K ααG G of group theory that a necessary condition for (15) to hold ααH −1 H is

if K is normal in A. And when K is not normal in A, [:ΓΓB ]|= S|. (16)

−1 FF−1 = αα. ααG G Moreover, using the theorems presented below together −1 H ααH with Lagrange’s theorem, it is possible to show that given

Brought to you by | De Gruyter / TCS Authenticated Download Date | 12/3/15 7:49 PM G.S. Chirikjian et al.: Decomposition of Sohncke space groups into Bieberbach and symmorphic parts 725

 a lattice translation subgroup Σ < ΓB such that Σ Γ and be viewed as the internal semi-direct product of the con- T n ΓS = ΣS < Γ, then another necessary condition for (15) to tinuous translation subgroup (n): = {(I, a)|a ∈ ℝ } and hold is the subgroup G(n): = {(A, 0)|A ∈ GL(n)} as TG G [:ΓΣ][=⋅ΓΣB :]||S . (17) Affn()==()nn() Tn() ()n . (18) The conditions (16) and (17) impose constraints on the Alternatively, we can write search for compatible Γ and Γ . Moreover, in the spirit B S = n (19) of simplicity, if Γ admits multiple decompositions of the Affn() R GL()n . form (15) (due to the existence of multiple Σ’s), we retain The distinction is that whereas T (n), G(n) < Aff(n), in the only the solution for which [Γ:Σ] is minimal. strictest sense the groups ℝn and GL(n) are not subgroups We note also that if [G:H] = 2 then H is simultaneously of Aff(n). Rather, they are groups isomorphic to T (n) and maximal, minimal index, and normal in G. And this facili- G(n), respectively. And so the symbol  means different tates our search, since many Bieberbach subgroups have things in (18) and (19). Equipped with the isomorphisms index two in their supergroup. T (n) ↔ ℝn and G(n) ↔ GL(n) this difference between In the remainder of this section a combination of internal and external direct products is small enough to known definitions and properties are reviewed and a new justify the double use of this symbol. And yet, without this theorem is presented to assist in the computation of these understanding, logical problems can arise. decompositions. For example, in mathematical crystallography we say

that a symmorphic space group, ΓP , with point group P can be written as (5). When doing so, this is the external Definitions involving internal products product in (19). If so then P is not a subgroup of ΓP . Rather, from the definition in (10) Suppose G is a group with identity element e and H and K are two normal subgroups of G. If P =={}00PP{( RRpp, )| ∈<} ΓP –– G = HK –– H ∩ K = {e} and we can write F Γ /T = P. then, G is called the (internal) direct product of H by K and P is denoted by G = H × K, and this product is symmetrical in In contrast to (11), this is an equality. Moreover, P  Γ but the sense that we can also write G = K × H. P < Γ . And so, for example, while it is possible to form an Suppose G is a group and H is a normal subgroup of G P external direct product of T and P to yield Γ = T  P, it is but K need not be. If P possible to construct the same thing as an internal direct –– G = HK product by simply multiplying T and P to yield Γ = TP. –– H ∩ K = {e} P We note that with the exception of P1, symmorphic groups are never direct products of T and P because then, G is called the (internal) semidirect product of H by

K and is denoted by G = H  K. Here H is normal in G and (,II t0γγ)( RR, )( , −=ttγγ)( , ()I − Rγγ). K ≅ G/H is called the complement of H in G. In a semidirect product, H  K, the complement K acts by conjugation on And since for nontrivial point group (I − Rγ)tγ cannot be

H. By definition of  this can also be written as G = K  H. equal to the zero translation for every possible Rγ in the

In general the order matters here, with the triangular part point group and tγ in the lattice, the subgroup P < ΓP is not of  or  pointing toward the normal subgroup. Note that closed under conjugation, and hence is not normal. GH== KHKG⇔=HK× . Since in the context of space groups the subgroups involved in every internal product can be identified with And so a direct product is always also a semi-direct product, simpler groups (by for example replacing group elements but not vice versa. In the case of a direct product, then for (A, 0) with A or (I, a) with a), it is often convenient to write every h ∈ H and every k ∈ K, hk = kh, whereas this is not true decompositions in terms of external products. However, in the case of semi-direct products. As a result, for a direct when doing so we must be careful to convert these exter- product the conjugation action of K on H is the trivial action. nal products to their equivalent internal description In contrast to internal (semi-)direct products, it is pos- before applying statements such as those at the beginning sible to define external products. For example, Aff(n) can of this section.

Brought to you by | De Gruyter / TCS Authenticated Download Date | 12/3/15 7:49 PM 726 G.S. Chirikjian et al.: Decomposition of Sohncke space groups into Bieberbach and symmorphic parts

We say the group G is decomposable if G is the direct representative from each right coset. As G is partitioned product of two normal subgroups, and semi-decompos- into right cosets, able if it is the semidirect product of a normal and non- ==F GK∪ gKKG\ (22) normal subgroup. ∈ gFKG\

and the analogous statement is true for left cosets: Identifying Γ and S subgroups B ==F Gg∪ KKGK/ . gF∈ GK\ For a Sohncke group, there are only three possibilities for the non-identity elements: They can be pure rotations, If N  G, then pure translations, or (nondegenerate) screw motions. That GN==FFN GG (23) is, every rigid-body transformation can be decomposed as NN (,RR tp)(=−, ()IR + dn) where the product of two subsets (not necessarily subgroups) A, B ⊂ G is defined as where n is the unit vector defining the axis of rotation of R, therefore satisfying Rn = n, p is a point on the screw AB =∈{|ab aA, bB∈ }. axis, and d is the amount of translation along the screw The product of subsets in a group inherits the associativity axis. Without loss of generality, we can always enforce the of the group. That is, given A, B, C ⊂ G, then (AB)C = A(BC). orthogonality condition p · n = 0. When d = 0, the motion is a pure rotation, though it will only be a rotation around Lemma 2.1: Let Σ  Γ be a lattice-translation subgroup the origin when in addition (I − R)p = 0. When R = I, the (i.e. Σ ≤ T), and let Σ < Γ , Γ < Γ where Γ is Bieberbach, motion is a pure translation, and we can take n = t/‖t‖. B S B Γ = ΣS = Σ  S, and S: = {0}  S with S < P (the point In the general case, simply by taking the dot product S group of Γ). Then the conditions of (I − R)p + dn = v(R) with n then gives

FFΓΓ= S mod Σ vn()Rd⋅=. (20) B (24) Σ Σ And so, evaluating (20) is a quick tool to identify what ΓΣ= FΓ S kinds of transformations populate fundamental domains B (25) Σ of finite coset spaces. If ℱ can be decomposed as a product of B and S Γ/Σ ΓΓ= S (26) subgroups as in (2), then some of those transformations B that are identified as screw will populate B and some of and those identified as rotations will populate S. We must first identify which combination of elements exhibit closure Γ Γ = B S. (27) ΣΣ with respect to the group operation of the original group, modulo Σ. are equivalent.

Proof: By definition, A = B mod Σ means that ΣA = ΣB and Main theorems used in computations the equality in (25) results from applying (23). Applying (23) again to (25) gives (26). Throughout this paper, we use the following results to do Also using condition (24), by an appropriate choice computations. ℱ ℱ of Γ/Σ every g ∈ Γ/Σ can be decomposed as g = bs where

bF∈ ΓΣ/ and s ∈ S. Therefore, using the associativity of Theorem 1: If N  H and N  Γ then B products of subsets in a group, every coset Σg = gΣ can be H Γ written as (Σb)s, which is the same as (27). □ H Γ ⇔ . (21) NN

Lemma 2.2: Let ΓB be a Bieberbach space group contain-  Proof: See [48]. □ ing lattice translation group Σ and let ΓS = ΣS = Σ S be a By definition, given a group G with K < G, a funda- symmorphic space group where S = {0}  S with S denot- mental domain FK\G ⊂ G is a set consisting of exactly one ing the point group of ΓS. Then

Brought to you by | De Gruyter / TCS Authenticated Download Date | 12/3/15 7:49 PM G.S. Chirikjian et al.: Decomposition of Sohncke space groups into Bieberbach and symmorphic parts 727

ΓΓ∩=Σ (28) BS But from Lemma 2.1, (30) gives Γ = ΓBS which together with Lemma 2.2 implies Γ  Γ. Then, since Γ ∩ S = {e}, it and B B  must be that Γ = ΓB S. Moreover, starting with this result Γ and working backwards, all of the steps are reversible. B ∩=S {}Σ . (29) Σ Using (27) and recognizing that (31) is equivalent to Γ Γ where e = (I, 0). B ΣΣ

Proof: (28) follows from the fact that elements of a (a result that could also be obtained using Theorem 1 and  Bieberbach group always have v(R) ≠ 0 except for v(I) = 0 ΓB Γ) together with (29) gives (34). Working in reverse whereas by an appropriate choice of origin, v(R) = 0 and evaluating (27) as a product of fundamental domains always. Therefore they can only share the subgroup Σ. mod T takes us back to (30). □ Using the modular law for intersections of subgroups, and rewriting (28) as Sample calculations ΓΣBB∩=()SSΣΓ()∩=Σ, and observing that ΓB ∩ S is a group that intersects Σ only Consider the nonsymmorphic space group Γ = P21212 and at e gives its maximal lattice translation group T = P1. Then, by the COSET function in the Bilbao Crystallographic server, ΓB ∩=Se{}. elements γ ∈FΓ can be visualized by their action on T FΓ t 3 Then since S ∩ Σ = {e}, for any choice of B containing x = [x, y, z] ∈ ℝ as Σ e we get γ ⋅∈x {(xy, , zx); (,−− yz, ); (1−+xy/2, +−1/2, z); F ∩=Se{}, +−+− ΓB (1xy/2, 1/2, z)}. Σ The elements {(x, y, z); (−x, −y, z)} are a conjugated which is equivalent to (29). □ version of those in the fundamental domain of P2/P1, and those in {(x, y, z); (−x + 1/2, y + 1/2, −z)} are a conjugated Theorem 2: Let Σ  Γ, and Σ < Γ , Γ < Γ satisfy the condi- B S version of those in P2 /P1. Moreover, each of these sub- tions in Lemma 2.1. Moreover, if 1 groups has index 2, and so both are normal. And so we FF= S mod Σ (30) can conclude that ΓΓB Σ Σ PP222 2 P2 11 =×1 then the following conditions are equivalent: PP11P1

−1 ssFΓΓ=∀Fs mod Σ ∈S (31) BB and so from Theorem 2, ΣΣ P2 ΓΓ (32) P222= P2 B 11 1 P1 ΓΓ= S (33) B where we have used the fact that a direct product is also and semi-direct (hence the conditions of the theorem are met); but a direct product in the quotient group does not carry Γ Γ = B S. (34) over to the parent group. ΣΣ As a second example, consider Γ = P42 and T = P1.   F Proof: Σ ΓB and Σ ΓS because normality of Σ in sub- Then elements of γ ∈ Γ are defined by groups of Γ containing Σ is inherited. Using condition (31) T and the normality of Σ, γ ⋅∈x {(xy,,zx);(,−− yz, );(,−+yx , z 1/2);

−−−− (,yx −+, z 1/2)}. 11FF11 ssΓΣB ==ssΓΓ()ssΣ ss BB ΣΣ Here again it is easy to see that a conjugated version ==ΣΓF . of ℱ is present, but the interpretation of the rest is ΓB B P2/P1 Σ not so clear. However, if we conjugate by α of the form

Brought to you by | De Gruyter / TCS Authenticated Download Date | 12/3/15 7:49 PM 728 G.S. Chirikjian et al.: Decomposition of Sohncke space groups into Bieberbach and symmorphic parts

ℱ in the section “Isomorphism vs. equality of fundamental 3. Use the information from Step 2 to compute Γ/ Σ . This domains” with a = 2, then the fundamental domain FΓ is equivalent in terms of affine conjugated fundamen- Σ F α . tal domains to Γ /1P This can be computed using for the resulting sublattice translation group Σ = αTα−1 is the Bilbao Server function COSETS. (It does not mat- defined as ter whether left or right cosets are computed because they are equivalent modulo P1, which must be normal (,xy , zx); (,−− yz, );(,−+yx , z 1/4); from the previous step); (,yx −+, zx1/4);( , yz, + 1/2); . 4. Evaluate the elements of ℱΓ Σ to determine whether −− +−+−+ / (,xy , zy1/2); (, xz, 3/4);(yx, , z 3/4) they are pure rotations or screw displacements using (20). Screw displacements and pure rotations are held This result is the same as doubling the basis vector as candidate elements of B and S, respectively. along the c-axis (as is given in the International Tables). 5. Construct all possible B and S subgroups and retain Within this P2/P1 is still visible, but now it becomes those pairs for which |B| · |S| = |Γ/Σ|; easier to see that P41/P1 is also present. And this has an 6. For those that satisfy 5, check if BS = Γ/Σ (or equiva- index of 2, indicating normality, and so we can write the FF lently ΓΣ//S = ΓΣ mod Σ ), and keep all of these; external semi-direct product B 7. Check whether B  Γ/Σ or S  Γ/Σ. If B  Γ/Σ, then go P2 to Step 9. Otherwise record and continue; P4421= P . P1 8. If no (B, S) pairs survive and if [Γ:Σ]/[ΓB:Σ] < [Γ:T] then increase n by 1 and go to Step 2. Otherwise continue;

9. Having stored all B, S pairs, find the names of ΓB and Γ Outline of decomposition S that gave rise to them. This can be achieved by comparing the number and nature of these elements algorithms for Sohncke groups (e.g. the value of d and rotation angle θ) against those parameters for the 8 nontrivial Bieberbach Sohncke We have approached the problem of semi-decomposition group types and 23 nontrivial symmorphic Sohncke of space groups from two different directions. These are group types. Alternatively, the information about called “top-down” and “bottom-up” as described below. these groups can be used by the Bilbao Server function IDENTIFY GROUP to find their names. After Top-down procedure names are identified in this way, end. Space groups have an infinite number of elements. To reduce to finite computations, in the top-down proce- We note that for all P groups except those with names start- dure we perform all calculations in a quotient of the form Γ ing P42 we have found through trial and error that it is pos- :/==ΓΣ ΣΓ\ where Σ ≤ T and Σ  Γ, and we seek finite Σ sible to take Σ = T = P1. That is, only a single loop through subgroups B, S < Γ/Σ that respectively have coset repre- the above procedure with n = 1 will suffice. Indeed, this is sentatives with no torsion and with only torsion such that how our manual search for such decompositions began. representatives of each coset in Γ/Σ can be written as a Coset representatives with respect to subgroup T com- product of representatives of cosets in B and S. Then using puted using the Bilbao Server function COSETS are easily the presented theorems, it becomes possible to make classified as torsion elements or torsion-free by evaluating statements about the decomposition of Γ itself. Below is (20). Those that are torsion free have d ≠ 0 and are kept our procedure for finding Σ and identifying B and S. This as candidates to populate B and those that have torsion procedure, which we have done mostly manually, is out- have d = 0 and are kept as candidates for S. We could visu- lined as an algorithm. ally inspect these candidates to identify different possi- 1. Set n = 1 and read off [Γ:T] = |P| from the International ble B and S subgroups. Since these will not necessarily Tables or the Bilbao Server; appear as they do in their standard form, we computed 2. Compute all subgroup paths from Γ to P1 of index screw parameters and rotation angles and axes to identify n · [Γ:T]. (For example, this can be done with the their type. This can all be done for relatively small [Γ:T] Bilbao Server SUBGROUPGRAPH function, which by hand.

returns the affine transformation α for each path). In cases such as P42 and several other groups with  α Retain only those for which P1 Γ . If n > 1 and no name “42” in it, it was not possible to identify B using the subgroup paths exists with P1  Γα, skip to 8; approach described in the previous paragraph, which led

Brought to you by | De Gruyter / TCS Authenticated Download Date | 12/3/15 7:49 PM G.S. Chirikjian et al.: Decomposition of Sohncke space groups into Bieberbach and symmorphic parts 729 us to use a coarser lattice Σ < T and to do the above com- 2. Whereas in the “top down” approach, T (or Σ) are putations with Γ/Σ instead of Γ/T. Here an affine transfor- guaranteed a priori to be common to both ΓS and ΓB, mation must be identified that can be used to conjugate there is no guarantee that this will be the case here. P1, simultaneously preserving S and coarsening the trans- That is, it is possible for lational sublattice. Using this coarser lattice, as in the ΣΓBB::=∩TT≠=ΣΓSS∩ , case of P42, will unveil B. In particular, we used the affine transformation with linear part diag[1, 1, 2] in conjunction in which case (35) will not hold. If Γ = Σ  S and Σ < Γ with the Bilbao Server COSET function. S S B B and Σ , Σ < T with Σ < Σ , then they are not compatible A similar approach is pursued for C, I, and F type S B B S for use in a semi-decomposition because then Γ ∩ Γ ≠ Σ . groups. In some cases such as C222 , it is possible to take B S S 1 But if Σ = Σ : = Σ, then we check all pairs for which Σ = T. But in most cases we have found that valid B and S B |B| · |S| = [Γ:Σ] and keep these as candidates. S subgroups can be identified when using Σ = P1 < T . We 3. The coset representatives for the remaining compat- call this approach “top down” because we start with the ible candidate pairs obtained in the previous step finest translation lattice shared by both Γ and Γ . When B S are then multiplied. If they reproduce ℱ , then they the search successfully concludes with Σ = T, then S and Γ/Σ represent a valid decomposition, and the procedure the B obtained in this way are both the largest ones pos- terminates here once all such decompositions are sible in their respective categories in the sense of having found. the minimal indices in Γ/Σ. The “bottom up” procedure described below is an Note that if a B or S group appears in the top node of alternative that seems to lend itself more easily to auto- the maximal subgroup graph then it will naturally have mation, and hence is useful for more difficult cases where minimal index within its type. That is, this copy will have [Γ:Σ] is large. smaller index in the parent group than a copy found further down in the maximal subgroup tree. If in addi- Bottom-up procedure tion it is normal (which we determine in the usual way When the top-down procedure described above becomes by conjugating elements of the subgroup by all elements too laborious, an entirely different approach can be taken. of the group and determining if the conjugates reside in If Γ , Γ < Γ share the same normal lattice translation sub- B S the subgroup), then this is a good candidate for a semi- group, i.e. Σ = T ∩ Γ = T ∩ Γ , then B S decomposition. If not, then working down the tree will ΓΣ==FFSSΓΓ= yield all minimal-index groups of a particular Bieberbach ΓΣBB//BSΓΣ or symmorphic type. and When decompositions cannot be found using the [:ΓΓ ]|==S|and [:ΓΓ][ΓΣ:]= ||B . BSB above algorithms In very few cases, the above procedures do not return Γ Σ = And if [ , ] |B| · |S| we will have that  results because there is no Σ Γ common to both ΓB and ⋅=ΓΓ ⋅ ΓΓ ||BS|| [:SB][ :]. (35) ΓS. In these rare cases (208, 214) we apply specialized methods as described in the Appendix. Assessing whether These statements hold regardless of whether or not Γ  Γ, B or not Σ is normal in Γ is easy. Given σ = (I, t ) ∈ Σ, then and can be used to assess potential compatibility of Γ and σ B define the lattice Λ: = {t |σ ∈ Σ}. Then Σ  Γ if and only if S in the sense of (3), as outlined below. σ Rγ · Λ = Λ for all Rγ ∈ P. 1. Search for all subgroup paths from Γ to ΓB and from

Γ to ΓS until [Γ:ΓB] ≤ |Smax| and [Γ:ΓS] ≤ 6. Here |Smax| is the order of the point group of the maximal symmorphic subgroup of Γ. If none exists use [Γ:T] in Table of semi-decomposable non- place of |Smax|, since there is no way for a proper subgroup of the point group to have order greater than symmorphic Sohncke space groups this. The number 6 is used for limiting the search for

ΓS because this is the largest possible value for |B|. Of the 230 space group types that are inequivalent under (We use the SUBGROUPGRAPH function in the Bilbao conjugation by Aff+(3), 13 types are Bieberbach groups. server to identify candidate Bieberbach and symmor- Included in this 13 is the group of pure lattice translations phic subgroups); of the primitive lattice, T = (P1)α. Since this translation

Brought to you by | De Gruyter / TCS Authenticated Download Date | 12/3/15 7:49 PM 730 G.S. Chirikjian et al.: Decomposition of Sohncke space groups into Bieberbach and symmorphic parts group is both Bieberbach and symmorphic and contains below and no |Ξ| > 1.6 Groups that we are not able to semi- no rotations, reflections, screws, or glides, we refer to it as decompose are 90, 208, 210, 214, and these are explained both a “trivially Bieberbach” and “trivially symmorphic” in greater detail in the Appendix. In the case of 90 and space group. 210 it is possible to write Γ = ΓBS with ΓB not normal in Γ.

In all cases it is possible to write Γ = ΓB′ΓS where multiple

choices exist for ΓB′ . In such cases, it is possible to find a Nontrivial Sohncke and non-Sohncke Σ < Γ such that7 Bieberbach types F ΣΓ\ ==BS′′ΞΣand\Γ BXS (36) The table below excludes T and indicates which of the Γ where B′ is a fundamental domain of B′ in Γ, Ξ is a remaining Bieberbach groups are also Sohncke. ΓB′ ∩T set of translations such that ΣΞ ≤ T, and S is as before. X is the group with elements that are cosets of the form ξΣ = Σξ Intl. Γ Sohncke? [ Γ :T ] B with ξ ∈ Ξ. When |Ξ| = 1, Γ is decomposed into a product

4 P21 Y 2 (semi-direct or otherwise) of ΓB and S. 7 Pc N 2 Decompositions of the form in (36) are generally not 9 Cc N 2 unique, and unlike the case when |Ξ| = 1, there is no clear 19 P212121 Y 4 criteria for stopping the search. Therefore, when |Ξ| ≠ 1 we 29 Pca21 N 4

33 Pna21 N 4 list examples to show that (36) is possible rather than to

76 P41 Y 4 attempt an exhaustive enumeration. 78 P4 Y 4 3 Using the frequencies of occurrence tabulated in [3], 144 P3 Y 3 1 altogether these groups that are not semi-decomposable 145 P3 Y 3 2 are only represented in the PDB less than one percent of 169 P61 Y 6 the time. 170 P65 Y 6  In constructing the table below, we seek ΓB Γ and

ΓS < Γ that enable the sorts of decompositions described

The quotient groups P212121/P1, Pca21/P1, and Pna21/P1 earlier. We start our search with small values of the can, respectively, be written as direct products of the product of indices form (P2 )α/P1 × (P2 )β/P1, (Pc)α/P1 × (P2 )β/P1, and (Pc)α/ 1 1 1 =⋅ β ΠΓ:[:]ΓΓBS[:Γ ]. P1 × (P21) /P1 for different conjugations α and β. Note that in mathematics books, 10 space group types When two combinations of Γ and Γ have the same are given including the trivial one consisting of only lattice B S minimal value of Π we list both next to each other in the translations. This is because they do not distinguish between table to the left of the | symbol, and those with larger the enantiomorphic pairs (P3 , P3 ), (P4 , P4 ), (P6 , P6 ). 1 2 1 3 1 5 values of Π are listed to the right of the | symbol. In all but As there are 230 isomorphism classes of space groups, one case, the complementing S is the same for decompo- showing all of the explicit calculations to decompose sitions corresponding to minimal-value of Π (which we them would require hundreds of pages of text. Instead call an “efficient” decomposition) and for those with here we only summarize the results for the Sohncke larger value of Π. In the one case when this happens, groups, giving enough information to retrace our calcula- C222 , another Γ is listed such that S′ is the complement tions using routines in the Bilbao Crystallographic Server 1 S′ of the less efficient Γ . When the answer to the normal- [45–47]. B′ ity question about S is different for the efficient and inef- ficient decompositions, it is listed to the right of | in the normality column. This only happens for one Sohncke Most Sohncke groups are semi- group, P6 22. decomposable 3

Of the 230 space group types, 65 are Sohncke. Of these 65, ℱ 6 Here Ξ consists of all pure translation elements of Σ\Γ and when 24 are symmorphic (including T) and eight are nontrivial Γ ΣΞ Γ ΣΓ , = BXS, where X =<. Bieberbach. Of the remaining 33 nonsymmorphic non- Σ ΣΣ Bieberbach Sohncke space groups, 29 are semi-decom-  7 Note that even if Σ is not normal in Γ, if Σ ΓB′ and ΣΞS = ΓS then  posable, which corresponds to a ‘Y’ in the B Γ/Σ column multiplying (36) on the left by Σ and using (22) gives Γ = ΓB′ΓS .

Brought to you by | De Gruyter / TCS Authenticated Download Date | 12/3/15 7:49 PM G.S. Chirikjian et al.: Decomposition of Sohncke space groups into Bieberbach and symmorphic parts 731

If we cannot find a Γ  Γ such that [Γ:Γ ] ≤ |S|, where the one with smallest possible index [Γ:Γ ]) has a F B B B ΓΓ/ B S is the point group of the the symmorphic subgroup that consists of not only point transformations, but pure

ΓS < Γ with minimal [Γ:ΓS], then we terminate the search. fractional translations. This happens when the largest

This strategy ensures that the calculation is finite. lattice translation subgroup of ΓB is smaller than that for  In the table that follows, we identify elements of a ΓS, i.e. ΣB < ΣS. And similarly, when there exists ΓS Γ and Bieberbach or symmorphic subgroup with the name given F contains Bieberbach group elements for a coarser ΓΓ/ S if we can find Γ′, Γ″ and affine transformations α, β such lattice than ΣS we write |Ξ| > 1 and N in the B column. In that cases when Σ = T ∩ ΓB is not normal in Γ, then Σ\Γ is not a

−1 group and it does not make sense to evaluate normality of FFΓΓ= αα′ B B and S, and so ‘N/A’ is listed in the table. Since these cases Σ T ′ B do not fit the semi-decomposition paradigm, this leads us and to state and prove the following theorem so as to include

−1 these cases in a decomposition that is more general than FFΓΓ= ββ′′ . S Γ = Γ S (but not as useful for our application). T ′′ B ΣS

In this case ΓB inherits the name of Γ′ and ΓS inherits the Theorem 3: Every Sohncke space group can be written as name of Γ″. a product of the form

ΓΓ= BSΓ .  Intl. Γ ΓB ΓS |Ξ| B, S Σ\Γ? Σ = T? Proof: There are several cases in the above table: 17 P222 P2 P2 1 Y, Y Y 1 1 (1), Σ  Γ and B  Γ/Σ with S < Γ/Σ; 18 P2 2 2 P2 |P2 2 2 P2 1 Y, Y Y 1 1 1 1 1 1 (2), Σ  Γ and S  Γ/Σ with B  Γ/Σ; 20 C2221 P21|P212121 C2|P2 1 Y, Y Y (3), Σ  Γ and ,  Γ/Σ; 24 I212121 P212121 P2 1 Y, N N B S   77 P42 P41, P43 P2 1 Y, N N (4), Σ Γ and B Γ/Σ with S ≠ F . ΓΓ/ B 80 I4 P4 , P4 P2 1 Y, N N 1 1 3   (5), Σ Γ and S Γ/Σ with FFΓΣ≠ ΓΓ . 90 P4212 P21 C222, P4 1 N, Y Y BB//S 91 P4 22 P4 P2 1 Y, N Y 1 1  (6), Σ is not normal in Γ and Σ\Γ = BXS with Σ ΓB and 92 P41212 P212121, P41 C2 1 Y, N Y ΓS = ΣΞS. 93 P4122 P41, P43 P222 1 Y, N N 94 P4 2 2 P2 2 2 , P4 , P4 C222 1 Y, N N 2 1 1 1 1 1 3 In Cases 1, 2, and 3, ΓΣ= F S. But since Σ = ΣΣ, and Σ  Γ, 95 P4 22 P4 P2 1 Y, N Y Γ 3 3 Σ 96 P4 2 2 P2 2 2 , P4 C2 1 Y, N Y 3 1 1 1 1 3 in all of these cases we can write 98 I4122 P212121, P41, P43 C222 1 Y, N N

151 P3112 P31 C2 1 Y, Y Y ΓΣ= ΣF S ΓΣB / 152 P3 21 P3 C2 1 Y, Y Y 1 1 ==()ΣΣF ()S ΓΓ. ΓΣB / BS 153 P3212 P32 C2 1 Y, N Y 154 P3 21 P3 C2 1 Y, N Y 2 2 The only difference between the cases is which of the sub- 171 P62 P32|P61 P2 1 Y, Y Y groups Γ or Γ are normal. In fact, as per Lemma 2.1 these 172 P64 P31|P65 P2 1 Y, Y Y B S

173 P63 P21|P61, P65 P3 1 Y, Y Y calculations do not require either of them to be normal.

178 P6122 P61 C2 1 Y, N Y But when it does exist, normality will be inherited from 179 P6 22 P6 C2 1 Y, N Y 5 5 the quotient groups according to the fifth column in the 180 P6 22 P3 |P6 C222 1 Y, N Y 2 2 1 table. 181 P6422 P31|P65 C222 1 Y, N Y

182 P6322 P21|P61, P65 P321, P312 1 Y, Y |N Y Case 4 is slightly different. In this case Γ  Γ but Σ < Σ . 198 P213 P212121 R3 1 Y, N Y B B S

199 I213 P212121 R3 1 Y, N N And so

208 P4232 P21 P23 4 N/A, N/A N ΓΣ= FF. B ΓΣBB//ΓΣSB 210 F4132 P212121, P41, P43 F23 1 N, Y N 212 P4 32 P2 2 2 R32 1 Y, N Y 3 1 1 1  But again using the fact that ΣB = ΣBΣB and ΣB Γ, we get 213 P4132 P212121 R32 1 Y, N Y 214 I4 32 P2 2 2 R32 2 Y, N N 1 1 1 1 ΓΣ==()FF()ΣΓΓ , BBΓΣBB//ΓΣSB BS

In the instance where |Ξ| > 1 and N appears in this table which follows from Lemma 2.1. Case 5 follows in the same in the S column, the largest normal Bieberbach group (i.e. way with the roles of B and S reversed.

Brought to you by | De Gruyter / TCS Authenticated Download Date | 12/3/15 7:49 PM 732 G.S. Chirikjian et al.: Decomposition of Sohncke space groups into Bieberbach and symmorphic parts

Case 6 follows because all Sohncke space groups can be written as products of Bieberbach and symmorphic subgroups. F ΓΣ==ΣΓ\ ΣΞBS □ We focus on the Sohncke case because that is what ==ΣΣBSΞΣ()BS()ΣΞ = ΓΓ. BS arises in the application that motivated us to study this subject. Moreover, the frequency of occurrence of Sohncke space groups populated by macromolecular crystals are Discussion predominantly those that admit such decompositions. The approach outlined for obtaining these decomposi- From the table, we see that 29 of the 33 nonsymmorphic tions can be applied to a number of non-Sohncke space Sohncke space groups can be decomposed into semi- groups as well, and we will pursue this in future work. direct products of Bieberbach subgroups and point subgroups. We also mention that in several cases in addition Acknowledgments: The authors would like to thank Prof. to being able to write Γ/Σ = B  S it is also the case that Bernard Shiffman and the anonymous reviewers for their Γ/Σ = B  S, and so Γ/Σ = B × S. This is true for groups very constructive feedback. with the following international numbers: 17, 18, 20, 151, 152, 171, 172, 173, 182. We note also that amongst the space groups that cannot be semi-decomposed, there is at least some product structure in the quotient group. For example space group A Additional examples of the

90 only has a normal copy of Bieberbach group P21 with decomposition procedure index 8, and so [Γ:ΓB] ≠ |S| where |S| = 4 for the largest subgroup of the point group, S, which has an associated Here additional examples of decompositions are provided. symmorphic subgroup of Γ. First, we show an example where decomposing Γ/T does The reason why the quotient in the case of 214 does not work, but Γ/Σ does. Next we show a case when no not decompose into two factors is that the only normal semi-decomposition is possible. Finally, we show a case

Bieberbach group with [Γ:ΓB] < [Γ:T] is ΓB = P212121 with where an additional (less efficient) semi-decomposition is

[Γ:ΓB] = 12. But the largest symmorphic subgroup of 214 is possible.

R32 which has |S| = [R32:TR] = 6. And so there is no way to make [Γ:ΓB] = |S|. And it is not reverse semi-decomposable because R32 is not a normal subgroup. Nevertheless, A.1 24, I212121

Theorem 3 applies and provides a simple tool to write I4132 as a product of affine-conjugated versions of P212121 and As another example, consider Γ = I212121. To determine T, R32 even in this case. we use the Bilbao function HERMANN which gives It should be noted that these four cases of 90, 208, −−1/21/2 00 210, and 214 that do not semi-decompose, respectively, α =−− represent 0.43, 0.03, 0.08, and 0.10 percent of all protein 1/21/2 11/4  crystal structures in the PDB [3]. In other words, more −1/21/2 00 than 99 percent of protein crystals occur in space groups that are either Bieberbach, symmorphic, or a semidirect (where the perfunctory bottom row in the affine transfor- product of a Bieberbach subgroup and a point subgroup. mation as a 4 × 4 matrix has been removed to save space.) The coset representatives in the decomposition of Γ with respect to T (in the basis of T) are given below:

F =−{( xy, , zy); (, −−xz, ); Conclusions Γ/T (1−+xy/2, −+1/2, −−xy++z 1/2); In this paper we introduced the idea that space groups (1yx+++/2, 1/2, xy−+z 1/2)}. lie along a spectrum with Bieberbach groups at one extreme and symmorphic groups at the other. Moreover, Here S = {(x, y, z); (−y, −x, −z)} is identical with F , CT2/ C we observe that 29 of the 33 nonsymmorphic Sohncke which is consistent with the fact that MAXSUB gives ℱ space groups can be decomposed into semi-direct prod- [I212121:C2] = 2. If Γ/T can be semi-decomposed, then it ucts of Bieberbach subgroups and point subgroups, and should be possible to find a Bieberbach subgroup with

Brought to you by | De Gruyter / TCS Authenticated Download Date | 12/3/15 7:49 PM G.S. Chirikjian et al.: Decomposition of Sohncke space groups into Bieberbach and symmorphic parts 733

[ΓB:P1] = 2. But the only Bieberbach group that could using Bilbao (or doing finite calculations in the factor satisfy this is P21, and MAXSUB does not list P21 as a group) we find that maximal subgroup. Therefore, there is no need to investi- I 222111/1PP= (211221 /1PP)(2/P1). gate the case when [I212121:TI] = 4 further, and so we investigate [I212121:Σ] = 8, which is the next smallest candidate. There is no guarantee that this decomposition is unique. When Σ = P1 < T the Bilbao COSET function (left and Using the bottom-up approach employing the Bilbao  right are the same so Σ Γ) gives MAXSUB function, we see that [I212121:P212121] = 2. and as [P2 2 2 :P1] = 4 and [P2:P1] = 2 we conclude that it is not F 1 1 1 IP222/1=−{( xy, , zx); (1+−/2, yz, + 1/2); 111 possible to find a decomposition with smaller values than (,−+xy 1/2, −+zx1/2); (1+−/2, yz+−1/2, ); this, and so our search terminates. (1xy++/2, 1/2, zx+−1/2); (, −+yz1/2, ); (1−+xy/2, , −−zx); (, yz, −+1/2)} A.2 90, P42 2 Here (x + 1/2, y + 1/2, z + 1/2) is a pure translation. Such will 1 only occur when Σ < T. In contrast, (–x + 1/2, –y, z + 1/2); Consider group no. 90, and use Γ = P42 2 and T = P1. When (–x, y + 1/2, –z+ 1/2); (x + 1/2, –y + 1/2, –z) are all screw 1 using the identity transformation, the Bilbao COSET func- motions with d = 1/2, and all of the rotation axes are mutu- tion gives ally orthogonal with an angle of rotation of π. So this is a clue that it may be related to P2 2 2 /P1. We verify this by F =−{( xy, , zx); (, −yz, ); 1 1 1 PP421 2/ 1 using the Bilbao COSET function to give (1−+yx/2, ++1/2, zy); (1/2, −+xz1/2, ); F (1−+xy/2, +−1/2, zx); (1+−/2, yz+−1/2, ); PP222/1=−{( xy, , zx); (1+−/2, yz, + 1/2); 111 (,yx , −−zy); (, −−xz, )} (,−+xy 1/2, −+zx1/2); (1+−/2, yz+−1/2, )}

This matches with four entries in F . In more This is convenient because we can immediately identify IP222111/1 complicated cases we might need to search for an affine S1 :{=−(,xy , zx); (, −−yz, ); (,yx , zy); (,−− xz, − )} ℱ transformation to relate a subgroup of Γ/Σ with a candidate Bieberbach group quotient by an appropriate trans- with a point group of order 4 (and index 2 in Γ). Therefore lation group. it is normal. Searching for every symmorphic space group In contrast, looking at the elements (–x, –y + 1/2, z); with point group of order 4, we find that this corresponds

(–x + 1/2, y, –z); (x, –y, –z + 1/2), we observe that they are to C222/TC. The complementing subgroup can be chosen all rotations because d = 0. But if we seek B and S such as F that their product reproduces IP222/1, then both cannot 111 B:{=−(,xy , zx); (1++/2, yz1/2, − )}. have order 4. And because B = 2 2 2 /P1 is of order 2 in 1 1 1 (In fact, there are four possible choices, but they all act in F when evaluated modulo P1, it is normal, and so IP222111/1 the same way as described below.) It is easy to check that we do not need to check that the conjugation action of Si this is a group. on F . For the nontrivial transformation in this group, R is a IP222111/1 t We see that we can define a two-element point group rotation around n = e2 by π. And v(R) = [1/2, 1/2, 0] . And so

(which fixes a point other than the origin) from the iden- v · n = 1/2. We conclude that B is isomorphic with P21/P1. tity and any one of (–x, –y + 1/2, z); (–x + 1/2, y, –z); (x, –y, To find affine transformations that will make the relation-

–z + 1/2). For example, ship between B and P21/P1, we solve the equation αg = g′α where g and g′ are respectively representatives taken from S1 :{=−(,xy , zx); (, −+yz1/2, )}, B and P21/P1. S2 :{=−(,xy , zx); (1+−/2, yz, )}, Clearly F = SB. Next we ask if B is normal. To PP4212/ 11 S3 :{=−(,xy , zx); (, yz, −+1/2)}. do this, we use the Bilbao COSET function with the information about α that has been computed. We find that left Abstractly, every two-element group is isomorphic, and and right cosets do not match modulo P1, and B is not so candidates for these are P2/P1 or C2/T . But concretely, C normal, and we can only find affine transformations such that α F α−1 = . Again examining left and right COSETS F= SB. iP2/Pi1 Si PP421 2/ 11

Brought to you by | De Gruyter / TCS Authenticated Download Date | 12/3/15 7:49 PM 734 G.S. Chirikjian et al.: Decomposition of Sohncke space groups into Bieberbach and symmorphic parts

The benefit of this top-down approach is that we need “reverse-semi-decomposed” in multiple ways, each with not worry about whether the two components correspond the pure rotation part being normal rather than the Bie- to space subgroups with a common lattice, since that is berbach part. As a consequence of this, and of Theorem 3, guaranteed a priori. But this does not provide a complete it is possible to write list of all possible decompositions. We therefore go to the P42 22==PP42PC222 Bilbao MAXSUB function, and look for other low-index 11 1 symmorphic and Bieberbach subgroups. We find that where it is understood that in the above different copies [P42 2:P4] = 2. Hence P4  P42 2. The same subroutine 1 1 of P2 are used. Moroever, from Lemma 2.1, this is a case provides a transformation, and we find (after moding out 1 where even though Γ is not normal, it is nevertheless pos- extraneous 1’s returned by Bilbao) that B sible to perform decompositions of the form Γ = ΓBS.

F −1 =−{( xy, , zx); (, −−yz, ); (,yx , zy); (, −xz, ); (4ααPP221 )/ 1 (1−+xy/2, +−1/2, zx); (1+−/2, yz−−1/2, ); (1yx+−/2, 1/2, −−zy); (1−−/2, xz−−1/2, )}. A.3 93, P4222

Using the Bilbao COSET function, and taking the top-

Here we see that the first four elements correspond to down approach, [P4222:P1] = 8 and : = P4/P1, and with (–x + 1/2, y + 1/2, –z) clearly visible, S2 F PP422/ 1 =−{( xy, , zx); (, −yz, ); we can define the complement B as before. And it is not 2 normal here either. Therefore, we conclude that (,−+yx , zy1/2); (, −+xz, 1/2); (,−−xy , zx); (, −−yz, );

F−1 = . (4ααPP22 )/ 1 SB2 1 (,yx , −+zy1/2); (,−− xz, −+1/2)}.

Though the quotient is “reverse-semi-decomposable” Of these, the Bieberbach subgroup is not normal (and no other =−− Bieberbach subgroup with [Γ:ΓB] < [Γ:T] is normal either. S {( xy, , zx); (, yz, ); F There is a copy of P212121 that has index 4, which we (,−−xy , zx); (, −−yz, )} = PP222/ 1 can identify using the HERMANN subroutine in the Bilbao server, and gradually increasing the index of candidate is a subgroup with [P4 22:P222] = 2 and hence it is normal. Bieberbach subgroups. But left and right cosets do not 2 But there is no way to construct a two-element subgroup match, and hence it is not normal. from the remaining elements, which appear to be P4 or Another route to index-4 P2 2 2 is by observing that 1 1 1 1 P4 transformations. And so we look for ways to conjugate F 3 none of the coset representatives in (4ααPP22 −1 )/ 1 given 1 so that all elements of these subgroups are present as above have translations in the z component. Then, using representatives in the coset decomposition. the affine transformation α with linear part diag[1, 1, 2] Using HERMANN routine in Bilbao with G = 93 and and zero translation in the COSET function in Bilbao H = 76 with index 4, we find that with can double the size of the quotient group, giving enough F room to fit both a copy of a conjugated PP222/1 and 111 1000 the same S as before, which is left unaffected by this =  affine conjugation. Moreover, S is normal in the resulting α 0100 16-element quotient group as well. And so we can write 0021/2  Γ/Σ = (C222/TC) (P212121/P1). But P212121 is not normal in F −1 =−{( xy, , zx); (, −yz, ); (4ααPP2 22 )/ 1 P4212, and because this quotient group has 16 elements (,−+yx , zy1/4); (, −+xz, 1/4); as opposed to the 8 of the original, we do not include this (,−−xy , zx−−1/2); (, yz, −−1/2); in the table. (,yx , −−zy1/4); (,−− xz, −−1/4); There is an affine-conjugated copy of P2 with 1 (,xy , zx+−1/2); (, −+yz, 1/2); α = diag[1, 1, 2] that is normal in P42 2 with index 8, but 1 − +−+ this index does not match the order of the largest point (,yx , zy3/4); (, xz, 3/4); subgroup, and so the compatibility conditions fail, and so (,−−xy , zx); (, −−yz, ); it cannot be used in a semi-decomposition. (,yx , −−zy3/4); (,−− xz, −−3/4)}. We therefore conclude that 90 cannot be efficiently semi-decomposed, but the quotient group can be Several subgroups can be visually identified:

Brought to you by | De Gruyter / TCS Authenticated Download Date | 12/3/15 7:49 PM F α =−{( xy, , zx); (, −−yz, ); (,yx , zy); (,−− xz, − ); (4PP22)/ 1 G.S. Chirikjian et al.: Decomposition of Sohncke space21 groups into Bieberbach and symmorphic parts 735 (1−+yx/2, ++1/2, zy1/4); (1+−/2, xz++1/2, 1/4); (1−+xy/2, +−1/2, zx++1/4);( 1/2, −+yz1/2, −+1/4); S =−{( xy, , zx); (, −yz, ); (,xy , zx+−1/2); (, −+yz, 1/2); F −+ ++ +−++ (,−−xy , zx); (, −−yz, )} = PP222/ 1 , (1yx/2, 1/2, zy3/4); (1/2, xz1/2, 3/4); (1−+xy/2, +−1/2, zx−+1/4); (1/2, −+yz1/2, −−1/4); B1 =−{( xy, , zy); (, xz, + 1/4); (,−−xy , zy+−1/2); (, xz, +=3/4)},F (,yx , −−zy1/2); (,−− xz, −−1/2)} PP4/1 1

B2 =−{( xy, , zy); (, xz, + 1/4); And (,−−xy , zy+−1/2); (, xz, +=3/4)} F . P 4/1 P 1 S =−{( xy, , zx); (, −−yz, ); (,yx , zy); (,−− xz, − )}

We find that = = F mod P1. And by conjugat- B1S B2S P222/P1 F looks like CT222/ ing each element of B by all elements of S and evaluat- C 1 Of the remainder of elements, we can construct ing mod P1, we see that B1 is closed under conjugation.

The same is true for B2. Hence, we find that B1 and B2 are B:{=−(,xy , zy); (1++/2, xz1/2, + 1/4); normal in F , and so −− ++−+ + PP422 2/ 1 (,xy , zy1/2); (1/2, xz1/2, 3/4)}. P422124= PP( 222 /1P )

F α , B is normal in (4PP22)/1 and so and 21 P4224= PC ( 222 /)T 21 1 C P422324= PP( 222 /1P )

And the same is true for P43: P422124= PC3 ( 222 /)TC

A.4 94, P42212 We note also that there are normal affine-conju-

gated copies of P222 and P212121 inside of P42212 with [P4 2 2, P1] = 8. When computing F using the [(P4 2 2)α:P222] = [(P4 2 2)β:P2 2 2 ] = 4. The difficulty with 2 1 PP42212/ 1 2 1 2 1 1 1 1 Bilbao COSET function, we find four coset reps that are P222 is that it has a different lattice than the Bieberbach pure rotations. subgroups with compatible orders, and so it cannot be

But also present are elements such as (–x + 1/2, y + 1/2, used to semi-decompose. P212121 does have a complement

–z + 1/2) and (x + 1/2, –y + 1/2, –z + 1/2), each of which are that is C222/TC (about a point that is not the origin when conjugated versions of elements of P21. But, comput- expressed in the basis of P212121 in the standard setting) ing all right coset reps for P42212/P21 in HERMANN, and giving computing the corresponding left cosets in COSET, we P42212 = PC222111 ( 222 /)TC . see that the P21 elements corresponding to this are not normal. In contrast, (–y + 1/2, x + 1/2, z + 1/2) and (y + 1/2, A.5 98, I4122 –x + 1/2, z + 1/2) which are in P41, and neither of which forms a 2-element group mod P1 with the identity. But P4 with index 4 is normal. [I4 22:P1] = 16 in standard when using 1 1 setting. Subgroup of pure rotations is order 4. So should 1000 be decomposable.  α = 0100  F IP422/ 1 = 0020 1 {( xy, , zx); (1−+/2, −+yz1/2, + 1/2);

α in Bilbao, [(P42212) , P1] = 16. (,−+yx 1/2, zy++1/4); (1/2, −+xz, 3/4); P4  (P4 2 2)α; [(P4 2 2)α, P4 ] = 4 and [(P4 2 2)α, 1 2 1 2 1 1 2 1 (1−+xy/2, , −+zx3/4); (, −+yz1/2, −+1/4); C222] = 4. ++−+ −−− In particular, (1yx/2, 1/2, zy1/2); (, xz, ); (1xy++/2, 1/2, zx+−1/2); (, −yz, ); ( −yx++1/2, , zy3/4); (, −+xz1/2, + 1/4); F α =−{( xy, , zx); (, −−yz, ); (,yx , zy); (,−− xz, − ); (4PP2122)/ 1 (1−+yx/2, ++1/2, zy1/4); (1+−/2, xz++1/2, 1/4); (,−+xy 1/2, −+zx1/4); (1+−/2, yz, −+3/4); (1−+xy/2, +−1/2, zx++1/4);( 1/2, −+yz1/2, −+1/4); (,yx , −−zy); (1+−/2, xz+−1/2, + 1/2)} (,xy , zx+−1/2); (, −+yz, 1/2); (1−+yx/2, ++1/2, zy3/4); (1+−/2, xz++1/2, 3/4); (1−+xy/2, +−1/2, zx−+1/4); (1/2, −+yz1/2,Brought −−1/ to4) you; by | De Gruyter / TCS (,yx , −−zy1/2); (,−− xz, −−1/2)} Authenticated Download Date | 12/3/15 7:49 PM 736 G.S. Chirikjian et al.: Decomposition of Sohncke space groups into Bieberbach and symmorphic parts

Of these, which is a pure rotation group. This is a case where we could decompose Γ/Σ (which S =−{( xy, , zy); (, −−xz, ); (,−−xy , zy); (, xz, − ); has 12 elements)

This is a conjugated version of F . ΓΣ/ = BS CT222/ C

B =−{( xy, , zy); (, xz++1/2, 1/4); where B = P61/Σ and (1−+xy/2, −+1/2, zy+−1/2); (1++/2, xz, 3/4)} S:{=−(,xy , zx); (, −=yz, )} PP2/ 1.

Similar less-efficient decompositions are possible for F −1 This is of the form α PP4/ 1α where α ∈ SE(3), and B is 1 P222 , C222 , P6 , P6 , P6 22, P6 22, P6 22. These are listed F 1 1 3 4 2 4 3 normal in IP422/ 1 , and so 1 in the table after the | symbol. In most cases they have the Γ I421124= PC( 222 /)TC . same type of complementing S as the B in the more efficient decomposition of the same group, and the answer to Similarly, by choosing different coset representatives to the question of whether S is normal in Σ\Γ is the same as define B we find well. When the answers differ, the answer for the less effi- I421324= PC(222 /)TC . cient decomposition is written to the right of the | symbol. It is also possible to identify elements of P2 2 2 in F 1 1 1 IP421 2/ 1 and to write A.7 208 and 210 I421 2 = PC222111 ( 222 /)TC . These two groups each have large symmorphic sub-

groups of index 2, indicating normality of ΓS. Since in a semi-decomposition all of the torsion must reside in S, if

A.6 171 P62 – A case that can be alternatively a compatible Bieberbach subgroup exists it must be the Γ Γ = decomposed with larger [Γ:Σ] case that [ : B] |S|. Each of the groups 208 and 210 have a symmorphic subgroup with point group of order 12 (195 F It is obvious from examining P62 that it is possible to write in 208, and 196 in 210). Moreover, in each of these two P 1 groups are Bieberbach groups of index 12 (76 and 78 in P6 = P3  (P2/P1), and in this case [Γ:Γ ] = [P6 :P3 ] = 2 2 2 B 2 2 208, and 76, 78 and 19 in 210). Checking these (e.g. by con- and [Γ :T] = [P3 :P ] = 3. B 2 1 jugating generators of each candidate Γ by all generators Here we show that it is possible to find other decom- B of Γ) none of these Bieberbach subgroups are normal. In positions where [Γ:Γ ] is the same but [Γ :Σ] is larger. In B′ B′ some cases there is a faster way to rule out Γ ’s that are not particular, [P6 :P6 ] = 2 and [P6 :P1] = 6 with B 2 1 1 normal, based on the discussion at the end of Section 6, 1000 which leads to the following theorem.  α = 0100. Theorem 4: If Σ = Γ ∩ T  Γ < Γ and Σ is not normal in 0020 B B Γ, then ΓB is not normal in Γ. The coset representatives from the two cosets are Proof: Every element of Γ can be written as γ = σb for {( xy, , zy); (,−− xy, z + 1/3); B B some σ ∈ Σ and b ∈FΓ . By construction, FΓ ∩=Te{} −+ −+ −− + B B (,xy xz, 2/3); (,xy , z 1/2); Σ Σ (,yx −+yz, +−5/6); (,xy xz, + 1/6)} where T is the minimal-index translation subgroup of Γ. Conjugating by an arbitrary γ ∈ Γ gives and −−11−−11 γγB γγ==σγbb()γσγγ()γ . {( −+xy, −+xz, 1/6); (,xy , z + 1/2); (,−−yx yz, +−5/6); (,xy xz, + 2/3); Conjugation does not change the nature of a rigid-body (,−−xy , zy); (, −+xy, z + 1/3)}. displacement, i.e. pure translations remain pure translations, pure rotations remain pure rotations, and screw From these we can construct displacements remain screw displacements with the same θ and d. Therefore, γσγ–1 must be a translation, but since F PP6/ 6 =−{( xy, , zx); (, −yz, )}, –1 21 Σ is not normal in Γ it must be that γσγ ∉ Σ for at least

Brought to you by | De Gruyter / TCS Authenticated Download Date | 12/3/15 7:49 PM G.S. Chirikjian et al.: Decomposition of Sohncke space groups into Bieberbach and symmorphic parts 737

–1 one γ ∈ Γ. Therefore, γγBγ cannot be decomposed as σ′b′. down the tree. To do this, we double the unit cell as before

But since every element of ΓB can be decomposed in this by choosing α = diag[1, 1, 2]. This results in 48-element –1 way, it must be that γγBγ ∉ ΓB and therefore ΓB cannot be coset spaces normal in Γ. □ (4PP32)/αα11≠ PP\( 432) . Note that whereas the proof of this theorem is specific 22 to space groups, it can be viewed as an example of the In other words, this is a case where Σ is not normal in Γ. contrapositive of the statement Nevertheless, within F α we can identify groups PP1\(42 32) ΣΓ, BB ΓΣ⇒∩ΓΓ Bx1 =−{( , yz, ); (1yx++/2, 1/2, z + 1/4); which is true in more general (abstract) settings. That is, (,−−xy , zy++1/2); (1/2, −+xz1/2, +=3/4)} F α (4P 1 ) the intersection of normal subgroups is also normal. P 1 Bx=+{( , yz, ); (1yx/2, −+1/2, z + 1/4); As a consequence of this theorem, if Σ = ΓB ∩ T is the 2 −− +−+++=F primitive translation group for ΓB, and ΓB < Γ, then if Σ is (,xy , zy1/2); (1/2, xz1/2, 3/4)} α (4P 3 ) P 1 not normal in Γ we need not even check if ΓB is normal, because it is guaranteed not to be. Therefore, in such and  cases it cannot be the case that Γ = ΓB S. F −1 Moreover, even if there were normal Bieberbach sub- S ==ααP 23 groups further down the subgroup tree, these would be P 1 irrelevant for such decompositions because there would {( xy, , zx); (,−− yz, ); (,xy −−, z); (,−−xy , zz); (2 , xy, 1/2); (2−−zx, , 1/2)y ; be no way to match [Γ:ΓB] and |S|. In searching for ΓB (2−−zx, , 1/2)yz; (2 , −−xy, 1/2); (,yz 2, 1/2)x ; and S such that Γ = ΓBS, the indices and orders of coset spaces must match even if ΓB is not normal in Γ because (,yz −−2, 1/2)xy; (,−− 2,zx 1/2); (,−−yz 2, 1/2)x }

ΓB ∩ S = e. The above mentioned groups have compatible indices and orders for this, and these are explored in the Unfortunately, subsections that follow. ΓΣ≠=BS Γ S. iBi And no other α was found that would enable such a A.7.1 Details of 208 decomposition. Searching further down the subgroup

graph cannot result in a ΓBS decomposition. We therefore

There are no normal Bieberbach subgroups in this group search for products of the form ΓBΓS. of an index compatible with the condition [Γ:ΓB] = |S|. The Choosing α = diag[1, 2, 1] we can identify inside of largest symmorphic subgroup has F α (which is not a group) three groups B, Ξ, S: PP1\(42 32) Sx=−{( , yz, ); (,xy −−, zx); (, yz, − ); F −1 S ==ααP 23 (,xy −−, zz); (, xy, ); (,zx −−, y); P 1 (,−−zx , yz); (,−− xy, ); (,yz , x); {( xy, , zx); (,−− yz, ); (,xy −−, z); F (,−−xy , zz); (, 1/2,xy 2); (,zx −−1/2, 2)y ; (,−−yz , xy); (, −−zx, ); (,−−yz , x)} = P 23 P 1 (,−−zx 1/2, 2)yz; (,−− 1/2,xy 2); (2yz, 1/2, x); It has index 2, and hence it is normal and the quotient is (2−−yz, 1/2, xy); (2 , −−1/2,zx ); (2−−yz, 1/2, x)} a group of order 2. The only Bieberbach group with |B| = 2 Ξ = {( xy, , zx); (, yz+=1/2, )} F α Γ = (1P ) is B P21. But none of the elements of P4232 are elements P 1 of P21 with the same lattice. For example, {(x, y, z); (y + 1/2, x + 1/2, –z + 1/2)} is a 2-element subgroup of F that and PP432 2/ 1 is not in S. Though it is isomorphic with P2 /P1, it is not B =−{( xy, , zx); (, yz+−1/2, )} = F . 1 P 21 equal to F . Rather, it is an element of C2 sharing the P 1 PP2/1 1 same lattice as P4 32 and so we can write 2 But the product of these three does not reproduce

 F α ∩ Ξ = Ξ ∩ = ∩ = C2 PP1\(432) because even though B S B S {e} P4322==PP32 32C . 2 2  we find that (ΞS) ∩ B ≠ {e}. TC Therefore we search deeper down the subgroup graph But our goal was not to decompose space groups into prod- (i.e. for larger [Γ:Σ]) and find that there are multiple affine ucts of symmorphic subgroups, and so we look further transformations with linear part

Brought to you by | De Gruyter / TCS Authenticated Download Date | 12/3/15 7:49 PM 738 G.S. Chirikjian et al.: Decomposition of Sohncke space groups into Bieberbach and symmorphic parts

110 Ξ SB = F α , since a necessary condition for the product 11 ΣΓ\  of subgroups to be a group is permutability. This results in A =−110  ΓΓ==ΓΓΓ . 002 BS11SB or a similarity-transformed version of this corresponding to permutations of coordinate axis names. We there- A.7.2 Details of 210 fore express Γα in the basis of Σ = P1 with α = (A, 0) (since translations are irrelevant to identifying Bieberbach sub- Using SUBGROUPGRAPH provides a number of transfor- groups in the top-down procedure, and making the trans- mations to generate ℱ of order 48. We choose this order lation zero retains the origin allows us to easily identify Σ\Γ because that is what is required to match |B| · |S|. The point rotations.) This makes it easy to identify the largest linear parts of all of the transformations are all equivalent S (corresponding to P23) as under relabeling of axes, and the translational part is set Sx=−{( , yz, )( xy, −−, zy)( , −−xz, )( yx, , −z); to zero so that it is easier to identify point transformations. (1−−/2xy1/2,++zx 1/21/2yz+−, 1/21xy+ /2 ); Therefore, we choose (1/2xy++1/2,zx −−1/21/2yz+−, 1/21xy/2 ); 1/21/2 00 (1/2xy+−1/2,zx −−1/21/2yz−−, 1/21xy+ /2 );  α =−1/21/2 00. (1−−/2xy1/2,−+zx 1/21/2yz−−, 1/21x / 2)y ; 0010 (1−+/2xy1/2,−−zx 1/21++/2yz, 1/21xy+ /2 ); (1/2xy−−1/2,zx 1/21−+/2yz, −−1/21xy/2 ); Then the resulting (1−+/2xy1/2,+−zx 1/21+−/2yz, −−1/21xy/2 ); FF ΣΓ\ = α (1/2xy−+1/2,zx 1/21−−/2yz, 1/21xy+ /2 )}. PF1\(41 32)

We also identify has the following subsets B =+{( xy, , zx); (1/2, −−yz, + 1/4)} 1 B1 =−{( xy, , zx); (1+−/2, yz++1/2, 1/2); (,−+xy 1/2, −+zx1/4); (1+−/2, yz, −+3/4)} F (which is a conjugated version of PP2/ 1 ), 1 Bx2 =−{( , yz, ); (,yx ++1/2, z 1/4); (1−+xy/2, −+1/2, zy++1/2); (1/2, −+xz, 3/4)} Ξ1 =+{( xy, , zx); (1/2, yz+ 1/2, ); Bx=−{( , yz, ); (,yx ++1/2, z 1/4); (,xy , zx++1/2); (1/2, yz++1/2, 1/2)} 3 (1−+x /2, −+yz1/2, +−1/2); (1yx++/2, , z 3/4)} and and Bx=−{( , yz, ); (1xy+−/2, ++1/2, z 1/2); 2 Sx=−{( , yz, ); (,xy −−, zy); (, xz, ); (,−−yx , −z); (,−+xy 1/2, −+zx3/4); (1+−/2, yz, −+1/4)} (1−−/2xy1/2,++zx 1/21/2yz+−, 1/21xy+ /2 ); (1−−/2xy1/2,−+zx 1/21/2yz−−, 1/21xy/2 ); (which is a conjugated version of F ) PP222111/1 (1−+/2xy1/2,−−zx 1/21++/2yz, 1/21xy+ /2 ); (1/2xy−−1/2,zx 1/21−+/2yz, −1/2xy− 1/2); Ξ2 =+{( xy, , zx); (, yz, 1/2)}. (1/2xy++1/2,zx −−1/21/2yz+−, 1/21xy/2 ); In this case8 (1/2xy+−1/2,zx −−1/21/2yz−−, 1/21xy+ /2 );

F α = BSΞ ΣΓ\ ii (1−+/2xy1/2,+−zx 1/21+−/2yz, −−1/21xy/2 ); (1/2xy−+1/2,zx 1/21−−/2yz, 1/21xy+ /2 ); for i = 1, 2. Whereas ΓS = ΣΞ1S, we find that ΓS = ΣΞ1S and so it cannot be used. But changing the order we find These respectively correspond to P212121, P41, P43, and F23. It can be shown that

8 Even though F α is not a group, equality is still assessed by eval- ΣΓ\ 3 F = BS uating the translational part modulo Z since multiplication on the ΣΓ\ i left by (I, z) ∈ Σ = P1 only has the effect of changing the coset repre-

F α = sentatives used to define ΣΓ\ by adding an arbitrary translation. for i 1, 2, 3 and so we could write

Brought to you by | De Gruyter / TCS Authenticated Download Date | 12/3/15 7:49 PM G.S. Chirikjian et al.: Decomposition of Sohncke space groups into Bieberbach and symmorphic parts 739

−1 ΓΣ==BSiBΓ S S = ββF . i RT32/ R

SUBGROUPGRAPH can also be used to identify P4 and even though none of the Γ ’s are normal with index 12. 1 Bi P212121 as subgroups of P4132 with the correct index with fundamental domains F of the correct order. Of these ΓB /T only P2 2 2 is normal. A.8 213, P4 32 and 212, P4 32 1 1 1 1 3 We note that using any α given by SUBGROUPGRAPH for P4 32 and P4 with index 6 gives Taking the top-down approach, we use COSET to compute 1 1 representatives of the cosets in P4 32/P1. The result is FF= F . 1 ΓΓ//TTBB′′ΓΓ/

[P4132:P1] = 24. Of these, we identify four elements to construct F , and we find that this is normal in For example, when α is a translation by [1/4, 0, 1/2]t, PP222111/1 F PP432/ 1. 1 F =−{( xy, , zx); (, −+yz, 1/2); ΓB′ /T We also find three representatives (x, y, z); (z, x, y); (,−+yx , zy1/4); (, −+xz, 3/4)} (y, z, x) that are clearly rotations that preserve the origin. However, these rotations all have the axis of rotation and we choose nt = [1, 1, 1] /3, and so any point on the line passing through the origin with this direction will be preserved by F =−{( xy, , zz); (3+−/4, xy+ 3/4, ); ΓΓ/ B′ them. Since the goal is to clearly separate point and Bie- (3−+yz/4, , −+xx3/4); (1−+/4, −+zy1/4, −+1/4); berbach transformations, we seek the translation to the (1zy+−/2, ++1/4, xy1/2); (1++/2, xz1/2, −+1/4)} point along this line that maximizes the number of entries that have v = 0. In particular, we find that when using which IDENTIFY GROUP identifies with R32/TR. And so,

1003/8 even though this ΓB′ is not normal, we can write Γ = ΓB′S  F α = 0103/8 by conjugating ΓΓ/ to make it a set of rotations around  B′ the origin, and calling this S. Even so, we do not list P4 in 00 13/8 1 the table because it is not normal, and is P212121 is normal, and hence the most useful for our application. in the Bilbao COSET function, F −1 = BS where (4ααPP32 )/ 1 1 From the above it is clear that the top-down and Sx= {( , yz, ); (,zx , yy); (, zx, ); bottom-up approaches provide the same results. And the (,−−yx , −−zx); (, −−zy, ); (,−−zy , −x)} final result is

R32 == and P431 2 PP222111 431 R 2. TR Bx=−{( , yz, ); (1xy−−/4, −+3/4, z 1/2); (3−−xy/4, +−1/2, z − 1/4); Group 212, P4 32 follows in a similar way, but with F −1 3 (1xy+−/2, −−1/4, z −=3/4)}.ααPP222/1 –1 t 111 α → α (corresponding to a translation by –[3/8, 3/8, 3/8] t instead of [3/8, 3/8, 3/8] ). P43 and P212121 as subgroups of

We note also that F −1 = BS′ where P4332 with the correct index with fundamental domains (4ααPP1 32 )/ 1 to decompose, but again only P2 2 2 is normal, leading to Bx′ =−{( , yz, ); (1yx−+/2, 3/4, z + 1/4); 1 1 1 (1−−xy/4, −−3/4, z + 1/2); R32 P433 22==PP11122 433 R 2. (1yx+−/4, −+1/2, z 3/4)} TR

and ΓB′ can be identified with P41. Using the Bilbao Server function IDENTIFY GROUP finds that ΓS = R32.

Alternatively, using the bottom-up approach we A.9 214, I4132 search for Bieberbach and symmorphic subgrouos respectively of order 4 and 6 (index 6 and 4) with SUB- [Γ:T] = 24 with minimal index symmorphic subgroup 155 GROUPGRAPH. This shows that R32 is the only symmor- (which has index 4 and point group of order 6). Bieberbach phic subgroup of P4132 with these properties, and so we subgroups with the compatible value of |B| = 4 are groups identify it with 19, 76, 78. However, these all have index 12 in group 214.

Brought to you by | De Gruyter / TCS Authenticated Download Date | 12/3/15 7:49 PM 740 G.S. Chirikjian et al.: Decomposition of Sohncke space groups into Bieberbach and symmorphic parts

Therefore, Γ/T cannot be decomposed into the form BS [4] M. G. Rossmann, The molecular replacement method. Acta F Crystallogr. A 1990, 46, 73. because [Γ:ΓB] > |S| and no Γ /T for these ΓB’s exists in B [5] H. M. Berman, T. Battistuz, T. N. Bhat, W. F. Bluhm, P. E. ℱ . Γ/T Bourne, K. Burkhardt, Z. Feng, G. L. Gilliland, L. Iype, S. Jain, The next finest lattice Σ is the lattice in the conven- P. Fagan, J. Marvin, D. Padilla, V. Ravichandran, B. Schneider, tional setting, with [Γ:Σ] = 48. Σ is normal in Γ. As with N. Thanki, H. Weissig, J. D. Westbrook, C. Zardecki, The protein 213, translating the origin by [3/8, 3/8, 3/8]t relative to that data bank. Acta Cryst. 2002, D58, 899. in the standard centering allows us to write [6] G. McColm, Prospects for mathematical crystallography. Acta Crystallogr. A Found Adv. 2014, 70, 95. FF==F BSΞ [7] M. Nespolo, Does mathematical crystallography still have a ΓΓBSΓ Σ Σ Σ role in the XXI century? Acta. Crystallogr. A 2008, 64, 96. [8] M. Nespolo, G. McColm, The wingspan of mathematical crystal- where in the case of 214, lography. Acta Crystallogr. A Found Adv. 2014, 70, 317. [9] E. Ascher, A. Janner, Algebraic aspects of crystallography. Helv. F ==Bx{( , yz, ); (1−−xy/4, −−3/4, z + 1/2); ΓB Phys. Acta. 1965, 38, 551. Σ [10] E. Ascher, A. Janner, Algebraic aspects of crystallography II. (3−−xy/4, +−1/2, z − 1/4); Commun. Math. Phys. 1968, 11, 138. (1xy+−/2, −−1/4, z − 3/4)} [11] L. Auslander, An account of the theory of crystallographic groups. Proc. Amer. Math. Soc. 1956, 16, 230. [12] D. S. Farley, I. J. Ortiz, Algebraic K-theory of Crystallographic and Groups: The Three-Dimensional Splitting Case, (Lecture Notes Sx= {( , yz, ); (,zx , yy); (, zx, ); in Mathematics), Springer, New York, 2014. [13] D. Fried, W. M. Goldman, Three-dimensional affine crystallo- −−−−−− −−− (,yx , zx); (, zy, ); (,zy , x)} graphic groups. Adv. Math. 1983, 47, 1. [14] L. S. Charlap, Bieberbach Groups and Flat Manifolds (Universi- and text), Springer, New York, 1986. [15] P. Engel, Geometric Crystallography: An Axiomatic Introduction Ξ =+{( xy, , zx); (1/2, yz++1/2, 1/2)} to Crystallography, Springer, New York, 1986. [16] T. Kobayashi, T. Yoshino, Compact Clifford-Klein forms of sym- where Γ = P2 2 2 and Γ = R32. B 1 1 1 S metric spaces–revisited. arXiv preprint (http://arxiv.org/pdf/ As can be seen by the translation subgroup {(x, y, z); math/0509543.pdf), 2005. (x + 1/2, y + 1/2, z + 1/2)}, the fundamental domain F is [17] R. C. Lyndon, Groups and Geometry (London Mathematical ΓS Σ Society Lecture Note Series), Cambridge University Press, not a valid S. As mentioned earlier, attempting to use a Cambridge, England, 1985. finer lattice to remove this translation also prevents any [18] J. M. Montesinos, Classical Tessellations and Three-Manifolds, normal Bieberbach groups from existing. And seeking a Springer-Verlag, Berlin, 1987. coarser Σ will only worsen the problem by resulting in [19] V. Nikulin, I. R. Shafarevich, Geometries and Groups, Universi- text, Springer, New York, 2002. more translational elements in FΓ . Therefore, a semi- S [20] J. Ratcliffe, Foundations of Hyperbolic Manifolds, Springer, Σ decomposition is not possible for group 214. Neverthe- New York, 2010. less by the reasoning in Theorem 3, it is the case from the [21] A. Szczepanski, Geometry of Crystallographic Groups (Algebra and Discrete Mathematics), World Scientific, above that Singapore, 2012. [22] W. Thurston, Three-dimensional manifolds, Kleinian groups I43112 = PR22211 32. and hyperbolic geometry. Bull. Amer. Math. Soc. 1982, 6, 357. [23] E. B. Vinberg, V. Minachin, Geometry II: Spaces of Constant Curvature (Encyclopaedia of Mathematical Sciences) Volume 2, Springer, New York, 1993. References [24] J. A. Wolf, Spaces of Constant Curvature, AMS Chelsea Publishing, Providence, RI, 2010. [1] G. S. Chirikjian, Mathematical aspects of molecular replace- [25] T. Sunada, Topological Crystallography with a View Towards ment. I. Algebraic properties of motion spaces. Acta Crystallogr. Discrete Geometric Analysis, Surveys and Tutorials in the A 2011, 67, 435. Applied Mathematical Sciences, Volume 6, Springer, New York, [2] G. S. Chirikjian, Y. Yan, Mathematical aspects of molecular 2012. replacement. II. Geometry of motion spaces. Acta Crystallogr. A [26] S. L. Altmann, Induced Representations in Crystals and Mol- 2012, 68, 208. ecules, Academic Press, San Diego, CA, 1978. [3] G. S. Chirikjian, S. Sajjadi, D. Toptygin, Y. Yan, Mathemati- [27] M. I. Aroyo, A. Kirov, C. Capillas, J. M. Perez-Mato, cal aspects of molecular replacement. III. Properties of space H. Wondratschek, Bilbao crystallographic server. II. Represen- groups preferred by proteins in the Protein Data Bank. Acta tations of crystallographic point groups and space groups. Crystallogr. A Found Adv. 2015, 71, 186. Acta Crystallogr. A 2006, 62, 115.

Brought to you by | De Gruyter / TCS Authenticated Download Date | 12/3/15 7:49 PM G.S. Chirikjian et al.: Decomposition of Sohncke space groups into Bieberbach and symmorphic parts 741

[28] C. Bradley, A. Cracknell, The Mathematical Theory of Symmetry [40] H. T. Stokes, D. M. Hatch, Isotropy Subgroups of The 230 in Solids: Representation Theory for Point Groups and Space Crystallographic Space Groups, Word Scientific, Singapore, Groups, Oxford University Press, Oxford, England, 2010. 1989. [29] T. Janssen, Crystallographic Groups, Elsevier, Amsterdam, 1973. [41] T. Hahn, International Tables for Crystallography, Volume A: [30] M. Boisen, G. V. Gibbs, Mathematical Crystallography (Reviews Space Group Symmetry A, International Union for Crystallogra- in Mineralogy), Volume 15, Mineralogical Society of America, phy, 2002. 1990. [42] H. Wondratschek, U. Müller, International Tables for Crystallog- [31] G. Burns, M. Glazer, Space Groups for Solid State Scientists, raphy, Volume A1: Symmetry relations between space groups, 3rd ed., Academic Press, San Diego, CA, 2013. International Union of Crystallography, 2008. [32] R. A. Evarestov, V. P. Smirnov, Site Symmetry in Crystals: Theory [43] W. Hantzsche, H. Wendt, Dreidimensionale euklidische Raum- and Applications, Springer Series in Solid-State Sciences, 1993. formen. Math. Ann. 1935, 110, 593. [33] M. M. Julian, Foundations of Crystallography with Computer [44] J. H. Conway, H. Burgiel, C. Goodman-Strauss, The Symmetries Applications, 2nd ed., CRC Press, Boca Raton, FL, 2014. of Things, AK Peters/CRC Press, Boca Raton, Florida, USA, [34] U. Müller, Symmetry relationships between crystal structures: 2008. applications of crystallographic group theory in crystal chemis- [45] M. I. Aroyo, J. M. Perez-Mato, C. Capillas, E. Kroumova, try. International Union of Crystallography, 2013. S. Ivantchev, G. Madariaga, A. Kirov, H. Wondratschek, Bilbao [35] M. Senechal, Crystalline Symmetries: An Informal Mathemati- crystallographic server: I. Databases and crystallographic cal Introduction, CRC Press, Boca Raton, FL, 1990. computing programs. Zeitschrift für Kristallographie 2006, [36] J. F. Cornwell, Group Theory in Physics, Volume 1: An Introduc- 221, 15. tion, (Techniques of Physics), Academic Press, New York, [46] S. Ivantchev, E. Kroumova, G. Madariaga, J. M. Perez-Mato, 1997. M. I. Aroyo, SUBGROUPGRAPH: a computer program for analy- [37] W. Miller Jr., Symmetry Groups and Their Applications, Volume sis of group-subgroup relations between space groups. J. Appl. 50, (Pure and Applied Mathematics Series), Academic Press, Crystallogr. 2000, 33, 1190. New York, 1972. [47] S. Ivantchev, E. Kroumova, M. I. Aroyo, J. M. Perez-Mato, [38] M. El-Batanouny, F. Wooten, Symmetry and Condensed Matter J. M. Igartua, G. Madariaga, H. Wondratschek, SUPERGROUPS: Physics: A Computational Approach, Cambridge University a computer program for the determination of the supergroups Press, Cambridge, UK, 2008. of the space groups. J. Appl. Crystallogr. 2002, 35, 511. [39] S. K. Kim, Group Theoretical Methods and Applications to [48] M. Senechal, Morphisms of crystallographic groups: Kernels Molecules and Crystals, Cambridge University Press, 1999. and images. J. Math. Phys. 1985, 26, 219.

Brought to you by | De Gruyter / TCS Authenticated Download Date | 12/3/15 7:49 PM