Skip to content
Licensed Unlicensed Requires Authentication Published by De Gruyter (O) December 1, 2015

Decomposition of Sohncke space groups into products of Bieberbach and symmorphic parts

  • Gregory S. Chirikjian EMAIL logo , Kushan Ratnayake and Sajdeh Sajjadi


Point groups consist of rotations, reflections, and roto-reflections and are foundational in crystallography. Symmorphic space groups are those that can be decomposed as a semi-direct product of pure translations and pure point subgroups. In contrast, Bieberbach groups consist of pure translations, screws, and glides. These “torsion-free” space groups are rarely mentioned as being a special class outside of the mathematics literature. Every space group can be thought of as lying along a spectrum with the symmorphic case at one extreme and Bieberbach space groups at the other. The remaining nonsymmorphic space groups lie somewhere in between. Many of these can be decomposed into semi-direct products of Bieberbach subgroups and point transformations. In particular, we show that those 3D Sohncke space groups most populated by macromolecular crystals obey such decompositions. We tabulate these decompositions for those Sohncke groups that admit such decompositions. This has implications to the study of packing arrangements in macromolecular crystals. We also observe that every Sohncke group can be written as a product of Bieberbach and symmorphic subgroups, and this has implications for new nomenclature for space groups.

Corresponding author: Gregory S. Chirikjian, Department of Mechanical Engineering, The Johns Hopkins University, 3400 N. Charles Street, Baltimore, MD 21218, USA, E-mail:


The authors would like to thank Prof. Bernard Shiffman and the anonymous reviewers for their very constructive feedback.

A Additional examples of the decomposition procedure

Here additional examples of decompositions are provided. First, we show an example where decomposing Γ/T does not work, but Γ/Σ does. Next we show a case when no semi-decomposition is possible. Finally, we show a case where an additional (less efficient) semi-decomposition is possible.

A.1 24, I212121

As another example, consider Γ=I212121. To determine T, we use the Bilbao function HERMANN which gives


(where the perfunctory bottom row in the affine transformation 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:


Here S={(x, y, z); (−y, −x, −z)} is identical with C2/TC, which is consistent with the fact that MAXSUB gives [I212121:C2]=2. If ℱΓ/T can be semi-decomposed, then it should be possible to find a Bieberbach subgroup with [ΓB :P1]=2. But the only Bieberbach group that could satisfy this is P21, and MAXSUB does not list P21 as a maximal subgroup. Therefore, there is no need to investigate the case when [I212121:TI ]=4 further, and so we investigate [I212121:Σ]=8, which is the next smallest candidate.

When Σ=P1<T the Bilbao COSET function (left and right are the same so Σ◁Γ) gives


Here (x+1/2, y+1/2, z+1/2) is a pure translation. Such will only occur when Σ<T. In contrast, (–x+1/2, –y, z+1/2); (–x, y+1/2, –z+1/2); (x+1/2, –y+1/2, –z) are all screw motions with d=1/2, and all of the rotation axes are mutually orthogonal with an angle of rotation of π. So this is a clue that it may be related to P212121/P1. We verify this by using the Bilbao COSET function to give


This matches with four entries in I212121/P1. In more complicated cases we might need to search for an affine transformation to relate a subgroup of ℱΓ/Σ with a candidate Bieberbach group quotient by an appropriate translation group.

In contrast, looking at the elements (–x, –y+1/2, z); (–x+1/2, y, –z); (x, –y, –z+1/2), we observe that they are all rotations because d=0. But if we seek 𝔹 and 𝕊 such that their product reproduces I212121/P1, then both cannot have order 4. And because 𝔹=212121/P1 is of order 2 in I212121/P1 when evaluated modulo P1, it is normal, and so we do not need to check that the conjugation action of 𝕊i on I212121/P1.

We see that we can define a two-element point group (which fixes a point other than the origin) from the identity and any one of (–x, –y+1/2, z); (–x+1/2, y, –z); (x, –y, –z+1/2). For example,


Abstractly, every two-element group is isomorphic, and so candidates for these are P2/P1 or C2/TC . But concretely, we can only find affine transformations such that αiP2/P1αi1=Si. Again examining left and right COSETS using Bilbao (or doing finite calculations in the factor group) we find that


There is no guarantee that this decomposition is unique.

Using the bottom-up approach employing the Bilbao MAXSUB function, we see that [I212121:P212121]=2. and as [P212121:P1]=4 and [P2:P1]=2 we conclude that it is not possible to find a decomposition with smaller values than this, and so our search terminates.

A.2 90, P4212

Consider group no. 90, and use Γ=P4212 and T=P1. When using the identity transformation, the Bilbao COSET function gives


This is convenient because we can immediately identify


with a point group of order 4 (and index 2 in Γ). Therefore it is normal. Searching for every symmorphic space group with point group of order 4, we find that this corresponds to C222/TC . The complementing subgroup can be chosen as


(In fact, there are four possible choices, but they all act in the same way as described below.) It is easy to check that this is a group.

For the nontrivial transformation in this group, R is a rotation around n=e2 by π. And v(R)=[1/2, 1/2, 0]t. And so v·n=1/2. We conclude that 𝔹 is isomorphic with P21/P1. To find affine transformations that will make the relationship between 𝔹 and P21/P1, we solve the equation αg=gα where g and g′ are respectively representatives taken from 𝔹 and P21/P1.

Clearly P4212/P1=S1B. Next we ask if 𝔹 is normal. To do this, we use the Bilbao COSET function with the information about α that has been computed. We find that left and right cosets do not match modulo P1, and 𝔹 is not normal, and


The benefit of this top-down approach is that we need not worry about whether the two components correspond to space subgroups with a common lattice, since that is guaranteed a priori. But this does not provide a complete list of all possible decompositions. We therefore go to the Bilbao MAXSUB function, and look for other low-index symmorphic and Bieberbach subgroups. We find that [P4212:P4]=2. Hence P4◁P4212. The same subroutine provides a transformation, and we find (after moding out extraneous 1’s returned by Bilbao) that


Here we see that the first four elements correspond to 𝕊2:=P4/P1, and with (–x+1/2, y+1/2, –z) clearly visible, we can define the complement 𝔹 as before. And it is not normal here either. Therefore, we conclude that


Though the quotient is “reverse-semi-decomposable” the Bieberbach subgroup is not normal (and no other Bieberbach subgroup with [Γ:ΓB ]<[Γ:T] is normal either.

There is a copy of P212121 that has index 4, which we can identify using the HERMANN subroutine in the Bilbao server, and gradually increasing the index of candidate Bieberbach subgroups. But left and right cosets do not match, and hence it is not normal.

Another route to index-4 P212121 is by observing that none of the coset representatives in (αP4212α1)/P1 given above have translations in the z component. Then, using the affine transformation α with linear part diag[1, 1, 2] and zero translation in the COSET function in Bilbao can double the size of the quotient group, giving enough room to fit both a copy of a conjugated P212121/P1 and the same 𝕊 as before, which is left unaffected by this affine conjugation. Moreover, 𝕊 is normal in the resulting 16-element quotient group as well. And so we can write

Γ/Σ=(C222/TC ) ⋊ (P212121/P1). But P212121 is not normal in P4212, and because this quotient group has 16 elements as opposed to the 8 of the original, we do not include this in the table.

There is an affine-conjugated copy of P21 with α=diag[1, 1, 2] that is normal in P4212 with index 8, but this index does not match the order of the largest point subgroup, and so the compatibility conditions fail, and so it cannot be used in a semi-decomposition.

We therefore conclude that 90 cannot be efficiently semi-decomposed, but the quotient group can be “reverse-semi-decomposed” in multiple ways, each with the pure rotation part being normal rather than the Bieberbach part. As a consequence of this, and of Theorem 3, it is possible to write


where it is understood that in the above different copies of P21 are used. Moroever, from Lemma 2.1, this is a case where even though ΓB is not normal, it is nevertheless possible to perform decompositions of the form Γ=ΓBS.

A.3 93,P4222

Using the Bilbao COSET function, and taking the top-down approach, [P4222:P1]=8 and


Of these,


is a subgroup with [P4222:P222]=2 and hence it is normal. But there is no way to construct a two-element subgroup from the remaining elements, which appear to be P41 or P43 transformations. And so we look for ways to conjugate so that all elements of these subgroups are present as representatives in the coset decomposition.

Using HERMANN routine in Bilbao with G=93 and H=76 with index 4, we find that with


Several subgroups can be visually identified:


We find that 𝔹1𝕊=𝔹2𝕊=FP222/P1 mod P1. And by conjugating each element of 𝔹1 by all elements of 𝕊 and evaluating mod P1, we see that 𝔹1 is closed under conjugation. The same is true for 𝔹2. Hence, we find that 𝔹1 and 𝔹2 are normal in P4222/P1, and so




A.4 94,P42212

[P42212, P1]=8. When computing P42212/P1 using the Bilbao COSET function, we find four coset reps that are pure rotations.

But also present are elements such as (–x+1/2, y+1/2, –z+1/2) and (x+1/2, –y+1/2, –z+1/2), each of which are conjugated versions of elements of P21. But, computing all right coset reps for P42212/P21 in HERMANN, and computing the corresponding left cosets in COSET, we 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, –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 when using


in Bilbao, [(P42212)α,P1]=16.

P41◁(P42212)α; [(P42212)α, P41]=4 and [(P42212)α, C222]=4.

In particular,




looks like C222/TC

Of the remainder of elements, we can construct


𝔹 is normal in (P42212)α/P1, and so


And the same is true for P43:


We note also that there are normal affine-conjugated copies of P222 and P212121 inside of P42212 with [(P42212)α:P222]=[(P42212)β:P212121]=4. The difficulty with P222 is that it has a different lattice than the Bieberbach subgroups with compatible orders, and so it cannot be used to semi-decompose. P212121 does have a complement that is C222/TC (about a point that is not the origin when expressed in the basis of P212121 in the standard setting) giving


A.5 98,I4122

P41 with index 4 is normal. [I4122:P1]=16 in standard setting. Subgroup of pure rotations is order 4. So should be decomposable.


Of these,


This is a conjugated version of C222/TC.


This is of the form αP41/P1α1 where αSE(3), and 𝔹 is normal in I4122/P1, and so


Similarly, by choosing different coset representatives to define 𝔹 we find


It is also possible to identify elements of P212121 in I4122/P1 and to write


A.6 171P62– A case that can be alternatively decomposed with larger [Γ:Σ]

It is obvious from examining P62P1 that it is possible to write P62=P32⋊ (P2/P1), and in this case [Γ:ΓB ]=[P62:P32]=2 and [ΓB :T]=[P32:P1]=3.

Here we show that it is possible to find other decompositions where [Γ:ΓB] is the same but [ΓB:Σ] is larger. In particular, [P62:P61]=2 and [P61:P1]=6 with


The coset representatives from the two cosets are




From these we can construct


which is a pure rotation group.

This is a case where we could decompose Γ/Σ (which has 12 elements)


where 𝔹=P61/Σ and


Similar less-efficient decompositions are possible for P2221, C2221, P63, P64, P6222, P6422, P6322. These are listed in the table after the | symbol. In most cases they have the same type of complementing S as the ΓB in the more efficient decomposition of the same group, and the answer to the question of whether 𝕊 is normal in Σ\Γ is the same as well. When the answers differ, the answer for the less efficient decomposition is written to the right of the | symbol.

A.7 208 and 210

These two groups each have large symmorphic subgroups of index 2, indicating normality of ΓS . Since in a semi-decomposition all of the torsion must reside in 𝕊, if a compatible Bieberbach subgroup exists it must be the case that [Γ:ΓB ]=|𝕊|. Each of the groups 208 and 210 have a symmorphic subgroup with point group of order 12 (195 in 208, and 196 in 210). Moreover, in each of these two groups are Bieberbach groups of index 12 (76 and 78 in 208, and 76, 78 and 19 in 210). Checking these (e.g. by conjugating generators of each candidate ΓB by all generators of Γ) none of these Bieberbach subgroups are normal. In some cases there is a faster way to rule out ΓB ’s that are not normal, based on the discussion at the end of Section 6, which leads to the following theorem.

Theorem 4: If Σ=ΓBT◁ΓB <Γ and Σ is not normal in Γ, then ΓB is not normal in Γ.

Proof: Every element of ΓB can be written as γB =σb for some σ∈Σ and bΓBΣ. By construction, ΓBΣT={e} where T is the minimal-index translation subgroup of Γ. Conjugating by an arbitrary γ∈Γ gives


Conjugation does not change the nature of a rigid-body displacement, i.e. pure translations remain pure translations, pure rotations remain pure rotations, and screw displacements remain screw displacements with the same θ and d. Therefore, γσγ–1 must be a translation, but since Σ is not normal in Γ it must be that γσγ–1∉Σ for at least one γ∈Γ. Therefore, γγBγ–1 cannot be decomposed as σb′. But since every element of ΓB can be decomposed in this way, it must be that γγBγ–1∉ΓB and therefore ΓB cannot be normal in Γ. □

Note that whereas the proof of this theorem is specific to space groups, it can be viewed as an example of the contrapositive of the statement


which is true in more general (abstract) settings. That is, the intersection of normal subgroups is also normal.

As a consequence of this theorem, if Σ=ΓBT is the primitive translation group for ΓB , and ΓB <Γ, then if Σ is not normal in Γ we need not even check if ΓB is normal, because it is guaranteed not to be. Therefore, in such cases it cannot be the case that Γ=ΓB ⋊𝕊.

Moreover, even if there were normal Bieberbach subgroups further down the subgroup tree, these would be irrelevant for such decompositions because there would be no way to match [Γ:ΓB ] and |𝕊|. In searching for ΓB and S such that Γ=ΓBS, the indices and orders of coset spaces must match even if ΓB is not normal in Γ because ΓBS=e. The above mentioned groups have compatible indices and orders for this, and these are explored in the subsections that follow.

A.7.1 Details of 208

There are no normal Bieberbach subgroups in this group of an index compatible with the condition [Γ:ΓB ]=|𝕊|. The largest symmorphic subgroup has


It has index 2, and hence it is normal and the quotient is a group of order 2. The only Bieberbach group with |𝔹|=2 is ΓB =P21. But none of the elements of P4232 are elements 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 P4232/P1 that is not in S. Though it is isomorphic with P21/P1, it is not equal to P21/P1. Rather, it is an element of C2 sharing the same lattice as P4232 and so we can write


But our goal was not to decompose space groups into products of symmorphic subgroups, and so we look further down the tree. To do this, we double the unit cell as before by choosing α=diag[1, 1, 2]. This results in 48-element coset spaces


In other words, this is a case where Σ is not normal in Γ. Nevertheless, within P1\(P4232)α we can identify groups






And no other α was found that would enable such a decomposition. Searching further down the subgroup graph cannot result in a ΓBS decomposition. We therefore search for products of the form ΓB ΓS .

Choosing α=diag[1, 2, 1] we can identify inside of P1\(P4232)α (which is not a group) three groups B, Ξ, S:




But the product of these three does not reproduce P1\(P4232)α because even though B∩Ξ=Ξ∩S=BS={e} we find that (ΞS)∩B≠{e}.

Therefore we search deeper down the subgroup graph (i.e. for larger [Γ:Σ]) and find that there are multiple affine transformations with linear part


or a similarity-transformed version of this corresponding to permutations of coordinate axis names. We therefore express Γα in the basis of Σ=P1 with α=(A, 0) (since translations are irrelevant to identifying Bieberbach subgroups in the top-down procedure, and making the translation zero retains the origin allows us to easily identify point rotations.) This makes it easy to identify the largest S (corresponding to P23) as


We also identify


(which is a conjugated version of P21/P1),




(which is a conjugated version of P212121/P1)


In this case[8]


for i=1, 2. Whereas ΓS =ΣΞ1S, we find that ΓS =ΣΞ1S and so it cannot be used. But changing the order we find Ξ1SB1=Σ\Γα, since a necessary condition for the product of subgroups to be a group is permutability. This results in Γ=ΓB1ΓS=ΓSΓB1.

A.7.2 Details of 210

Using SUBGROUPGRAPH provides a number of transformations to generate ℱΣ\Γ of order 48. We choose this order because that is what is required to match |B|·|S|. The linear parts of all of the transformations are all equivalent under relabeling of axes, and the translational part is set to zero so that it is easier to identify point transformations. Therefore, we choose


Then the resulting


has the following subsets




These respectively correspond to P212121, P41, P43, and F23. It can be shown that


for i=1, 2, 3 and so we could write


even though none of the ΓBi’s are normal with index 12.

A.8 213,P4132 and 212,P4332

Taking the top-down approach, we use COSET to compute representatives of the cosets in P4132/P1. The result is [P4132:P1]=24. Of these, we identify four elements to construct P212121/P1, and we find that this is normal in P4132/P1.

We also find three representatives (x, y, z); (z, x, y); (y, z, x) that are clearly rotations that preserve the origin. However, these rotations all have the axis of rotation nt=[1,1,1]/3, and so any point on the line passing through the origin with this direction will be preserved by them. Since the goal is to clearly separate point and Bieberbach transformations, we seek the translation to the point along this line that maximizes the number of entries that have v=0. In particular, we find that when using


in the Bilbao COSET function, (αP4132α1)/P1=BS where




We note also that (αP4132α1)/P1=BS where


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 search for Bieberbach and symmorphic subgrouos respectively of order 4 and 6 (index 6 and 4) with SUBGROUPGRAPH. This shows that R32 is the only symmorphic subgroup of P4132 with these properties, and so we identify it with


SUBGROUPGRAPH can also be used to identify P41 and P212121 as subgroups of P4132 with the correct index with fundamental domains ΓB/T of the correct order. Of these only P212121 is normal.

We note that using any α given by SUBGROUPGRAPH for P4132 and P41 with index 6 gives


For example, when α is a translation by [1/4, 0, 1/2]t,


and we choose


which IDENTIFY GROUP identifies with R32/TR . And so, even though this ΓB is not normal, we can write Γ=ΓBS by conjugating Γ/ΓB to make it a set of rotations around the origin, and calling this S. Even so, we do not list P41 in the table because it is not normal, and is P212121 is normal, and hence the most useful for our application.

From the above it is clear that the top-down and bottom-up approaches provide the same results. And the final result is


Group 212, P4332 follows in a similar way, but with αα–1 (corresponding to a translation by –[3/8, 3/8, 3/8]t instead of [3/8, 3/8, 3/8]t). P43 and P212121 as subgroups of P4332 with the correct index with fundamental domains to decompose, but again only P212121 is normal, leading to


A.9 214,I4132

[Γ:T]=24 with minimal index symmorphic subgroup 155 (which has index 4 and point group of order 6). Bieberbach subgroups with the compatible value of |𝔹|=4 are groups 19, 76, 78. However, these all have index 12 in group 214. Therefore, Γ/T cannot be decomposed into the form 𝔹𝕊 because [Γ:ΓB ]>|𝕊| and no ΓB/T for these ΓB ’s exists in ℱΓ/T .

The next finest lattice Σ is the lattice in the conventional setting, with [Γ:Σ]=48. Σ is normal in Γ. As with 213, translating the origin by [3/8, 3/8, 3/8]t relative to that in the standard centering allows us to write


where in the case of 214,






where ΓB =P212121 and ΓS =R32.

As can be seen by the translation subgroup {(x, y, z); (x+1/2, y+1/2, z+1/2)}, the fundamental domain ΓSΣ is not a valid S. As mentioned earlier, attempting to use a finer lattice to remove this translation also prevents any normal Bieberbach groups from existing. And seeking a coarser Σ will only worsen the problem by resulting in more translational elements in ΓSΣ. Therefore, a semi-decomposition is not possible for group 214. Nevertheless by the reasoning in Theorem 3, it is the case from the above that



[1] G. S. Chirikjian, Mathematical aspects of molecular replacement. I. Algebraic properties of motion spaces. Acta Crystallogr. A2011, 67, 435.10.1107/S0108767311021003Search in Google Scholar PubMed PubMed Central

[2] G. S. Chirikjian, Y. Yan, Mathematical aspects of molecular replacement. II. Geometry of motion spaces. Acta Crystallogr. A2012, 68, 208.10.1107/S010876731105118XSearch in Google Scholar PubMed

[3] G. S. Chirikjian, S. Sajjadi, D. Toptygin, Y. Yan, Mathematical aspects of molecular replacement. III. Properties of space groups preferred by proteins in the Protein Data Bank. Acta Crystallogr. AFound Adv. 2015, 71, 186.Search in Google Scholar

[4] M. G. Rossmann, The molecular replacement method. Acta Crystallogr. A1990, 46, 73.10.1107/S0108767389009815Search in Google Scholar

[5] H. M. Berman, T. Battistuz, T. N. Bhat, W. F. Bluhm, P. E. Bourne, K. Burkhardt, Z. Feng, G. L. Gilliland, L. Iype, S. Jain, P. Fagan, J. Marvin, D. Padilla, V. Ravichandran, B. Schneider, N. Thanki, H. Weissig, J. D. Westbrook, C. Zardecki, The protein data bank. Acta Cryst.2002, D58, 899.Search in Google Scholar

[6] G. McColm, Prospects for mathematical crystallography. Acta Crystallogr. AFound Adv. 2014, 70, 95.Search in Google Scholar

[7] M. Nespolo, Does mathematical crystallography still have a role in the XXI century? Acta. Crystallogr. A2008, 64, 96.10.1107/S0108767307044625Search in Google Scholar PubMed

[8] M. Nespolo, G. McColm, The wingspan of mathematical crystallography. Acta Crystallogr. A Found Adv.2014, 70, 317.Search in Google Scholar

[9] E. Ascher, A. Janner, Algebraic aspects of crystallography. Helv. Phys. Acta. 1965, 38, 551.Search in Google Scholar

[10] E. Ascher, A. Janner, Algebraic aspects of crystallography II. Commun. Math. Phys. 1968, 11, 138.Search in Google Scholar

[11] L. Auslander, An account of the theory of crystallographic groups. Proc. Amer. Math. Soc.1956, 16, 230.Search in Google Scholar

[12] D. S. Farley, I. J. Ortiz, Algebraic K-theory of Crystallographic Groups: The Three-Dimensional Splitting Case, (Lecture Notes in Mathematics), Springer, New York, 2014.10.1007/978-3-319-08153-3Search in Google Scholar

[13] D. Fried, W. M. Goldman, Three-dimensional affine crystallographic groups. Adv. Math.1983, 47, 1.Search in Google Scholar

[14] L. S. Charlap, Bieberbach Groups and Flat Manifolds (Universitext), Springer, New York, 1986.10.1007/978-1-4613-8687-2Search in Google Scholar

[15] P. Engel, Geometric Crystallography: An Axiomatic Introduction to Crystallography, Springer, New York, 1986.10.1007/978-94-009-4760-3Search in Google Scholar

[16] T. Kobayashi, T. Yoshino, Compact Clifford-Klein forms of symmetric spaces–revisited. arXiv preprint (, 2005.10.4310/PAMQ.2005.v1.n3.a6Search in Google Scholar

[17] R. C. Lyndon, Groups and Geometry (London Mathematical Society Lecture Note Series), Cambridge University Press, Cambridge, England, 1985.Search in Google Scholar

[18] J. M. Montesinos, Classical Tessellations and Three-Manifolds, Springer-Verlag, Berlin, 1987.Search in Google Scholar

[19] V. Nikulin, I. R. Shafarevich, Geometries and Groups, Universitext, Springer, New York, 2002.Search in Google Scholar

[20] J. Ratcliffe, Foundations of Hyperbolic Manifolds, Springer, New York, 2010.Search in Google Scholar

[21] A. Szczepanski, Geometry of Crystallographic Groups (Algebra and Discrete Mathematics), World Scientific, Singapore, 2012.10.1142/8519Search in Google Scholar

[22] W. Thurston, Three-dimensional manifolds, Kleinian groups and hyperbolic geometry. Bull. Amer. Math. Soc.1982, 6, 357.Search in Google Scholar

[23] E. B. Vinberg, V. Minachin, Geometry II: Spaces of Constant Curvature (Encyclopaedia of Mathematical Sciences) Volume 2, Springer, New York, 1993.10.1007/978-3-662-02901-5Search in Google Scholar

[24] J. A. Wolf, Spaces of Constant Curvature, AMS Chelsea Publishing, Providence, RI, 2010.10.1090/chel/372Search in Google Scholar

[25] T. Sunada, Topological Crystallography with a View Towards Discrete Geometric Analysis, Surveys and Tutorials in the Applied Mathematical Sciences, Volume 6, Springer, New York, 2012.10.1007/978-4-431-54177-6_6Search in Google Scholar

[26] S. L. Altmann, Induced Representations in Crystals and Molecules, Academic Press, San Diego, CA, 1978.Search in Google Scholar

[27] M. I. Aroyo, A. Kirov, C. Capillas, J. M. Perez-Mato, H. Wondratschek, Bilbao crystallographic server. II. Representations of crystallographic point groups and space groups. Acta Crystallogr. A2006, 62, 115.10.1107/S0108767305040286Search in Google Scholar PubMed

[28] C. Bradley, A. Cracknell, The Mathematical Theory of Symmetry in Solids: Representation Theory for Point Groups and Space Groups, Oxford University Press, Oxford, England, 2010.Search in Google Scholar

[29] T. Janssen, Crystallographic Groups, Elsevier, Amsterdam, 1973.Search in Google Scholar

[30] M. Boisen, G. V. Gibbs, Mathematical Crystallography (Reviews in Mineralogy), Volume 15, Mineralogical Society of America, 1990.10.1515/9781501508912Search in Google Scholar

[31] G. Burns, M. Glazer, Space Groups for Solid State Scientists, 3rd ed., Academic Press, San Diego, CA, 2013.Search in Google Scholar

[32] R. A. Evarestov, V. P. Smirnov, Site Symmetry in Crystals: Theory and Applications, Springer Series in Solid-State Sciences, 1993.10.1007/978-3-642-97442-7Search in Google Scholar

[33] M. M. Julian, Foundations of Crystallography with Computer Applications, 2nd ed., CRC Press, Boca Raton, FL, 2014.Search in Google Scholar

[34] U. Müller, Symmetry relationships between crystal structures: applications of crystallographic group theory in crystal chemistry. International Union of Crystallography, 2013.10.1093/acprof:oso/9780199669950.001.0001Search in Google Scholar

[35] M. Senechal, Crystalline Symmetries: An Informal Mathematical Introduction, CRC Press, Boca Raton, FL, 1990.Search in Google Scholar

[36] J. F. Cornwell, Group Theory in Physics, Volume 1: An Introduction, (Techniques of Physics), Academic Press, New York, 1997.Search in Google Scholar

[37] W. Miller Jr., Symmetry Groups and Their Applications, Volume 50, (Pure and Applied Mathematics Series), Academic Press, New York, 1972.Search in Google Scholar

[38] M. El-Batanouny, F. Wooten, Symmetry and Condensed Matter Physics: A Computational Approach, Cambridge University Press, Cambridge, UK, 2008.10.1017/CBO9780511755736Search in Google Scholar

[39] S. K. Kim, Group Theoretical Methods and Applications to Molecules and Crystals, Cambridge University Press, 1999.10.1017/CBO9780511534867Search in Google Scholar

[40] H. T. Stokes, D. M. Hatch, Isotropy Subgroups of The 230 Crystallographic Space Groups, Word Scientific, Singapore, 1989.10.1142/0751Search in Google Scholar

[41] T. Hahn, International Tables for Crystallography, Volume A: Space Group Symmetry A, International Union for Crystallography, 2002.Search in Google Scholar

[42] H. Wondratschek, U. Müller, International Tables for Crystallography, Volume A1: Symmetry relations between space groups, International Union of Crystallography, 2008.Search in Google Scholar

[43] W. Hantzsche, H. Wendt, Dreidimensionale euklidische Raumformen. Math. Ann.1935, 110, 593.Search in Google Scholar

[44] J. H. Conway, H. Burgiel, C. Goodman-Strauss, The Symmetries of Things, AK Peters/CRC Press, Boca Raton, Florida, USA, 2008.Search in Google Scholar

[45] M. I. Aroyo, J. M. Perez-Mato, C. Capillas, E. Kroumova, S. Ivantchev, G. Madariaga, A. Kirov, H. Wondratschek, Bilbao crystallographic server: I. Databases and crystallographic computing programs. Zeitschrift für Kristallographie2006, 221, 15.10.1524/zkri.2006.221.1.15Search in Google Scholar

[46] S. Ivantchev, E. Kroumova, G. Madariaga, J. M. Perez-Mato, M. I. Aroyo, SUBGROUPGRAPH: a computer program for analysis of group-subgroup relations between space groups. J. Appl. Crystallogr.2000, 33, 1190.Search in Google Scholar

[47] S. Ivantchev, E. Kroumova, M. I. Aroyo, J. M. Perez-Mato, J. M. Igartua, G. Madariaga, H. Wondratschek, SUPERGROUPS: a computer program for the determination of the supergroups of the space groups. J. Appl. Crystallogr.2002, 35, 511.Search in Google Scholar

[48] M. Senechal, Morphisms of crystallographic groups: Kernels and images. J. Math. Phys.1985, 26, 219.Search in Google Scholar

Received: 2015-5-21
Accepted: 2015-10-16
Published Online: 2015-12-1
Published in Print: 2015-12-1

©2015 by De Gruyter

Downloaded on 22.9.2023 from
Scroll to top button