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\subsection{point groups}

A point group is a group of symmetry operations leaving at least one point unchanged, i.e. there exists a point $x$ such that $\forall \sigma \in G, \sigma x = x$.  These requirements are very general, admitting rotation groups of all orders, and inversions, and, of course, combinations of these operations.  The point groups are represented by matricies of determinant $\pm 1$ and are therefore the subgroups of $O(n)$.

\begin{theorem}
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In two dimensions, a rotation group of order $n$ (that is, consisting of rotations of $2\pi k/n$ radians for integers $k$) is denoted as the cyclic group $C_n$.  If we include inversion (improper rotations) then we generate the dihedral group $D_n$, the group of symmetries of a regular $n$-gon.  Similarly, the point groups in three dimensions are the symmetry groups of regular polyhedra, with and without allowing inversion.

\subsection{lattice groups}

A \emph{lattice} is a point set exhibiting translational symmetry.  In the plane we can represent this translational symmetry by two nonparallel basis vectors $\alpha$ and $\beta$, such that for $x, y \in X$ there are unique $a, b \in Z $ such that $y = a\alpha + b\beta$.

\subsection{the crystallographic restriction}

Any pattern repeating in space, such as a tesselation of the plane or a crystal in space, exists on such a lattice.  Therefore we are very interested in lattice symmetries.  In particular this brings about the \emph{crystallographic restriction}.

If a particular point set exhibits rotational symmetry about a point $x$ and it also exhibits translational symmetry by a vector $\alpha$ then it must also exhibit rotational symmetry about $x' = x + \alpha$.   It turns out that only rotations of order 1, 2, 3, 4, and 6 are possible in a uniform tiling of space.  This is known as the crystallographic restriction.

As we may also utilize the inversion operation, \emph{improper rotations} should also be included.  Thus the possible point-group symmetries on a lattice are $C_{1}, C_{2}, C_{3}, C_{4}, C_{6},  D_{1}, D_{2}, D_{3}, D_{4}, D_{6}$.

The crystallographic restriction may be proved by contradiction.  Suppose a wallpaper group has rotations of order $q$ and let $a$ be the shortest nonzero vector in L, the lattice group.  Then let $A$ be a rotation by an angle $2\pi/q$.  If $q > 6$ or if $q = 5$ then $Aa$ is shorter than A, violating the hypothesis concerning $a$.  The only remaining possibilities are $q \in \{1, 2, 3, 4, 6\}$.

\section{Space groups}

The space groups are the result of combining a point group with a lattice group.  Two dimensional space groups are called ``wallpaper groups'' since they represent every possible symmetry system of regular patterns tiling the plane; three dimensional space groups are called the ``crystallographic group.''  There are seventeen space groups in two dimensions, 230 in three dimensions, and 4901 in four dimensions.

Employing the trick of ``homogeneous coordinates'' we can represent an affine transformation in $n$ dimensions as an $n+1$ dimensional square matrix.  In this system point is represented as a column vector consisting of its $n$ coordinates augmented with a dummy coordinate with value $1$.   

In this way we can represent each basis vector of a translation group as a matrix, as well as a rotation matrix and the inversion matrix.  Thus we have the ingredients needed to generate a space group in any dimension.  The space groups are exactly those groups generated in this manner.  
We see immediately as well that any space group will have translation and rotation subgroups.  The existence of the subgroup of translations is known as Bieberbach's theorem.

As a regular point set fills space, the translational subgroup must be infinite.  Furthermore, translations are obviously commutative, so this subgroup is Abelian.  Senechal notes that the existence of this infinite commutative subgroup may be used to define the space groups.

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