View source for Crystallographic group

A discrete group of motions of an <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c027/c027190/c0271901.png" />-dimensional Euclidean space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c027/c027190/c0271902.png" /> having a bounded [[Fundamental domain|fundamental domain]]. Two crystallographic groups are said to be equivalent if they are conjugate in the group of affine transformations of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c027/c027190/c0271903.png" />.

The origin of the theory of crystallographic groups is connected with the study of the symmetry of ornaments <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c027/c027190/c0271904.png" /> and the structure of crystals <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c027/c027190/c0271905.png" />. A classification of all planar (two-dimensional) and spatial (three-dimensional) crystallographic groups was achieved at the end of the 19th century by E.S. Fedorov and somewhat later by A. Schoenflies (see [[#References|[2]]], [[#References|[3]]], and also [[#References|[6]]], [[#References|[7]]], [[#References|[9]]]). Up to equivalence, there are 17 planar and 219 spatial crystallographic groups; if, however, the spatial groups are considered up to conjugacy with respect to orientation-preserving affine transformations, their number is 230. In 1910, L. Bieberbach investigated crystallographic groups of arbitrary dimension . In particular, he proved the following theorems:

1) Any <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c027/c027190/c0271906.png" />-dimensional crystallographic group <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c027/c027190/c0271907.png" /> contains <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c027/c027190/c0271908.png" /> linearly independent parallel translations; the group <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c027/c027190/c0271909.png" /> of linear parts of the transformations from <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c027/c027190/c02719010.png" /> is finite. (For <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c027/c027190/c02719011.png" /> this was proved in [[#References|[3]]].)

2) Two crystallographic groups are equivalent if and only if they are isomorphic as abstract groups.

3) For any <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c027/c027190/c02719012.png" />, there are — up to equivalence — only finitely many <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c027/c027190/c02719013.png" />-dimensional crystallographic groups (this is a solution to Hilbert's 18th problem).

Theorem 1 yields the following description of the structure of crystallographic groups as abstract groups. Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c027/c027190/c02719014.png" /> be the set of all parallel translations in a crystallographic group <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c027/c027190/c02719015.png" />. Then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c027/c027190/c02719016.png" /> is a normal subgroup of finite index, isomorphic to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c027/c027190/c02719017.png" />, and is its own centralizer in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c027/c027190/c02719018.png" />. The existence of a normal subgroup <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c027/c027190/c02719019.png" /> of an abstract group <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c027/c027190/c02719020.png" /> possessing these properties is a sufficient condition for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c027/c027190/c02719021.png" /> to be isomorphic to a crystallographic group [[#References|[7]]].

The group <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c027/c027190/c02719022.png" /> of linear parts of a crystallographic group <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c027/c027190/c02719023.png" /> preserves the lattice <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c027/c027190/c02719024.png" />; in other words, relative to a basis of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c027/c027190/c02719025.png" /> the transformations in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c027/c027190/c02719026.png" /> are represented by matrices with integer entries.

In order to specify a crystallographic group <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c027/c027190/c02719027.png" /> it is necessary to specify — in addition to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c027/c027190/c02719028.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c027/c027190/c02719029.png" /> — a vector <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c027/c027190/c02719030.png" /> for each <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c027/c027190/c02719031.png" /> such that the transformation

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c027/c027190/c02719032.png" /></td> </tr></table>

belongs to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c027/c027190/c02719033.png" />. The vector <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c027/c027190/c02719034.png" /> is defined up to addition by a vector from <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c027/c027190/c02719035.png" />. The mapping

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c027/c027190/c02719036.png" /></td> </tr></table>

is a one-dimensional cocycle on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c027/c027190/c02719037.png" /> with values in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c027/c027190/c02719038.png" />, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c027/c027190/c02719039.png" /> is the vector space associated with <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c027/c027190/c02719040.png" />.

Any triple <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c027/c027190/c02719041.png" />, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c027/c027190/c02719042.png" /> is a finite linear group, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c027/c027190/c02719043.png" /> is a <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c027/c027190/c02719044.png" />-invariant lattice and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c027/c027190/c02719045.png" /> a one-dimensional cocycle on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c027/c027190/c02719046.png" /> with values in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c027/c027190/c02719047.png" />, corresponds as just described with some crystallographic group. Under this correspondence, two triples <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c027/c027190/c02719048.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c027/c027190/c02719049.png" />, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c027/c027190/c02719050.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c027/c027190/c02719051.png" /> are cohomologous cocycles, correspond to equivalent crystallographic groups. To the zero cohomology class corresponds the split (or symmorphic) crystallographic group, which, relative to a suitable choice of an origin, is the set of all transformations

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c027/c027190/c02719052.png" /></td> </tr></table>

where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c027/c027190/c02719053.png" />.

In matrix interpretation, the description of all <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c027/c027190/c02719054.png" />-dimensional crystallographic groups reduces to the description of all finite groups of square matrices of order <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c027/c027190/c02719055.png" /> with integer entries (up to conjugacy in the group <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c027/c027190/c02719056.png" />) and, for each such group <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c027/c027190/c02719057.png" />, to the computation of the cohomology group <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c027/c027190/c02719058.png" />.

Two cohomology classes define equivalent crystallographic groups if and only if they are transformed into one another by the normalizer of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c027/c027190/c02719059.png" /> in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c027/c027190/c02719060.png" />. Bieberbach's theorem 2 and a result due to H. Zassenhaus [[#References|[7]]] imply that the natural homomorphism

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c027/c027190/c02719061.png" /></td> </tr></table>

is an isomorphism. This is readily deduced from the exact cohomology sequence of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c027/c027190/c02719062.png" />.

Two crystallographic groups belong to the same class (arithmetical class) if their groups of linear parts are conjugate in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c027/c027190/c02719063.png" /> (in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c027/c027190/c02719064.png" />). For <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c027/c027190/c02719065.png" /> there are 32 classes and 73 arithmetical classes of crystallographic groups.

Among the finite groups of matrices with integer entries one can single out the lattice symmetry groups, i.e. the groups of all orthogonal transformations that preserve some fixed lattice in a vector space (and are presented relative to a basis of this lattice). In 1848, A. Bravais determined all possible <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c027/c027190/c02719066.png" />-dimensional lattice symmetry groups and accordingly divided all <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c027/c027190/c02719067.png" />-dimensional lattices into 14 types (known as Bravais types). The subgroups of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c027/c027190/c02719068.png" /> that are lattice symmetry groups are called Bravais subgroups.

The Bravais subgroups may also be interpreted as stabilizer subgroups for the natural action of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c027/c027190/c02719069.png" /> on the set of positive-definite quadratic forms in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c027/c027190/c02719070.png" /> variables. They may therefore be determined using reduction theory (see [[#References|[11]]] and [[Quadratic forms, reduction of|Quadratic forms, reduction of]]). Every maximal finite subgroup of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c027/c027190/c02719071.png" /> is a Bravais subgroup (but the converse is not true).

The following table lists the number of finite subgroups in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c027/c027190/c02719072.png" /> (up to conjugacy).''''''<table border="0" cellpadding="0" cellspacing="0" style="background-color:black;"> <tr><td> <table border="0" cellspacing="1" cellpadding="4" style="background-color:black;"> <tbody> <tr> <td colname="1" style="background-color:white;" colspan="1"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c027/c027190/c02719073.png" /></td> <td colname="2" style="background-color:white;" colspan="1">Number of maximal finite subgroups</td> <td colname="3" style="background-color:white;" colspan="1">Number of Bravais subgroups</td> <td colname="4" style="background-color:white;" colspan="1">Number of finite subgroups</td> </tr> <tr> <td colname="1" style="background-color:white;" colspan="1">1</td> <td colname="2" style="background-color:white;" colspan="1">1</td> <td colname="3" style="background-color:white;" colspan="1">1</td> <td colname="4" style="background-color:white;" colspan="1">2</td> </tr> <tr> <td colname="1" style="background-color:white;" colspan="1">2</td> <td colname="2" style="background-color:white;" colspan="1">2</td> <td colname="3" style="background-color:white;" colspan="1">5</td> <td colname="4" style="background-color:white;" colspan="1">13</td> </tr> <tr> <td colname="1" style="background-color:white;" colspan="1">3</td> <td colname="2" style="background-color:white;" colspan="1">4</td> <td colname="3" style="background-color:white;" colspan="1">14</td> <td colname="4" style="background-color:white;" colspan="1">73</td> </tr> <tr> <td colname="1" style="background-color:white;" colspan="1">4</td> <td colname="2" style="background-color:white;" colspan="1">9</td> <td colname="3" style="background-color:white;" colspan="1">64</td> <td colname="4" style="background-color:white;" colspan="1">710</td> </tr> <tr> <td colname="1" style="background-color:white;" colspan="1">5</td> <td colname="2" style="background-color:white;" colspan="1">17</td> <td colname="3" style="background-color:white;" colspan="1">?</td> <td colname="4" style="background-color:white;" colspan="1">?</td> </tr> </tbody> </table>

</td></tr> </table>

An intersection of Bravais subgroups is again a Bravais subgroup. The smallest Bravais subgroup <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c027/c027190/c02719074.png" /> containing the group <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c027/c027190/c02719075.png" /> of linear parts of a crystallographic group <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c027/c027190/c02719076.png" />, up to conjugacy in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c027/c027190/c02719077.png" /> (in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c027/c027190/c02719078.png" />), is known as the geometrical (arithmetical) holohedry of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c027/c027190/c02719079.png" />. If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c027/c027190/c02719080.png" /> is a crystallographic group in general position, in the sense that there is no affine transformation mapping it onto a crystallographic group whose lattice of parallel translations has lower symmetry, then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c027/c027190/c02719081.png" /> is the lattice symmetry group of parallel translations of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c027/c027190/c02719082.png" />. Two crystallographic groups belong to the same syngony (Bravais type) if their geometrical (arithmetical) holohedries coincide. For <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c027/c027190/c02719083.png" /> there are 7 syngonies and 14 Bravais types of crystallographic groups.

==Linear representations of crystallographic groups.==
The irreducible finite-dimensional complex linear representations of a crystallographic group <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c027/c027190/c02719084.png" /> are described as follows. Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c027/c027190/c02719085.png" /> be some character (a homomorphism into the multiplicative group of complex numbers) of the group <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c027/c027190/c02719086.png" />, put

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c027/c027190/c02719087.png" /></td> </tr></table>

and let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c027/c027190/c02719088.png" /> be an irreducible representation of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c027/c027190/c02719089.png" /> such that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c027/c027190/c02719090.png" />, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c027/c027190/c02719091.png" />. Then the representation of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c027/c027190/c02719092.png" /> induced by the representation <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c027/c027190/c02719093.png" /> of its subgroup <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c027/c027190/c02719094.png" /> (see [[Induced representation|Induced representation]]) is irreducible. All irreducible representations of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c027/c027190/c02719095.png" /> are obtained in this way (see [[#References|[9]]], [[#References|[10]]]).

====References====
<table><TR><TD valign="top">[1a]</TD> <TD valign="top"> A. Bravais, "Abhandlung über die systeme von regelmässig auf seiner Ebene oder Raum vertheilten Punkten" , Teubner (1897)</TD></TR><TR><TD valign="top">[1b]</TD> <TD valign="top"> A. Bravais, "Abhandlung über symmetrische Polyeder" , Teubner (1890)</TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top"> E.S. Fedorov, "The symmetry and structure of crystals. Fundamental works" , Moscow (1949) pp. 111–255 (In Russian)</TD></TR><TR><TD valign="top">[3]</TD> <TD valign="top"> A. Schoenflies, "Kristallsysteme und Kristallstruktur" , Teubner (1891)</TD></TR><TR><TD valign="top">[4a]</TD> <TD valign="top"> L. Bieberbach, "Ueber die Bewegungsgruppen der Euklidischen Räume I" ''Math. Ann.'' , '''70''' (1911) pp. 297–336</TD></TR><TR><TD valign="top">[4b]</TD> <TD valign="top"> L. Bieberbach, "Ueber die Bewegungsgruppen der Euklidischen Räume II" ''Math. Ann.'' , '''72''' (1912) pp. 400–412</TD></TR><TR><TD valign="top">[5]</TD> <TD valign="top"> B. Delone, N. Padurov, A. Aleksandrov, "Mathematical foundation for the lattice analysis of crystals" , Moscow-Leningrad (1934) (In Russian)</TD></TR><TR><TD valign="top">[6]</TD> <TD valign="top"> A.V. Shubnikov, "An atlas of crystallographic symmetry groups" , Moscow-Leningrad (1946) (In Russian)</TD></TR><TR><TD valign="top">[7]</TD> <TD valign="top"> H. Zassenhaus, "Ueber ein Algorithmus zur Bestimmung der Raumgruppen" ''Comm. Math. Helv.'' , '''21''' (1948) pp. 117–141</TD></TR><TR><TD valign="top">[8]</TD> <TD valign="top"> A.I. Mal'tsev, "Classical algebra" , ''Selected works'' , '''1''' , Moscow (1976) pp. 371–375 (In Russian) {{MR|}} {{ZBL|0422.01024}} </TD></TR><TR><TD valign="top">[9]</TD> <TD valign="top"> G.Ya. Lyubarskii, "Anwendung der Gruppentheorie in der Physik" , Deutsch. Verlag Wissenschaft. (1962) (Translated from Russian)</TD></TR><TR><TD valign="top">[10]</TD> <TD valign="top"> D.K. Faddeev, "Tables of the fundamental unitary representations of Fedorov groups" , Moscow-Leningrad (1961) (In Russian) {{MR|158025}} {{ZBL|}} </TD></TR><TR><TD valign="top">[11]</TD> <TD valign="top"> B.N. Delone, R.V. Galiulin, M.I. Shtrogrin, "On the Bravais types of lattices" ''J. Soviet Math.'' , '''4''' : 1 (1975) pp. 79–156 ''Itogi Nauki i Tekhn. Sovrem. Probl. Mat.'' , '''2''' (1973) pp. 119–254 {{MR|}} {{ZBL|0334.50005}} </TD></TR></table>



====Comments====
Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c027/c027190/c02719096.png" /> be the group of affine transformations of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c027/c027190/c02719097.png" />. The translations in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c027/c027190/c02719098.png" /> form a normal subgroup <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c027/c027190/c02719099.png" /> and the quotient <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c027/c027190/c027190100.png" /> is the group of linear transformations. Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c027/c027190/c027190101.png" /> be a subgroup of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c027/c027190/c027190102.png" />. Then the translations in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c027/c027190/c027190103.png" /> form a normal subgroup <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c027/c027190/c027190104.png" />; the quotient <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c027/c027190/c027190105.png" /> is often called the point group of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c027/c027190/c027190106.png" />; this is called the group of linear parts above.

Fedorov obtained his classification results during 1885–1889, whereas Schoenflies obtained a classification around 1891. The correct list of 230 groups was found only after comparing the lists of Fedorov and Schoenflies (see [[#References|[a1]]] for historical and other remarks). The book [[#References|[a2]]] by L. Sohnke was known to both.

For maximal finite subgroups of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c027/c027190/c027190107.png" /> for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c027/c027190/c027190108.png" /> one may consult [[#References|[a3]]].

====References====
<table><TR><TD valign="top">[a1]</TD> <TD valign="top"> R.L.E. Schwartzenberger, "<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c027/c027190/c027190109.png" />-dimensional crystallography" , Pitman (1980)</TD></TR><TR><TD valign="top">[a2]</TD> <TD valign="top"> L. Sohnke, "Entwicklung einer Theorie der Kristallstruktur" , Teubner (1879)</TD></TR><TR><TD valign="top">[a3]</TD> <TD valign="top"> W. Plesken, M. Post, "On maximal finite irreducible subgroups of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c027/c027190/c027190110.png" /> I, II" ''Math. Computation'' , '''31''' (1977) pp. 536–551; 552–573</TD></TR><TR><TD valign="top">[a4]</TD> <TD valign="top"> M. Klemm, "Symmetrien von Ornamenten und Kristallen" , Springer (1982) {{MR|0663004}} {{ZBL|0482.20034}} </TD></TR><TR><TD valign="top">[a5]</TD> <TD valign="top"> J.J. Burkhardt, "Die Bewegungsgruppen der Kristallographie" , Birkhäuser (1947)</TD></TR></table>