Bieberbach's theorems

The Bieberbach theorems show that there are only a finite number of space groups in each dimension. In 1910, Ludwig Bieberbach solved the 18th of David Hilbert's 23 mathematical problems .

Crystallographic Groups

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The isometric group of -dimensional Euclidean space is the group ${\displaystyle Isom(\mathbb {R} ^{n})}$ ${\displaystyle n}$ ${\displaystyle \mathbb {R} ^{n}}$

${\displaystyle Isom(\mathbb {R} ^{n})=O(n)\ltimes \mathbb {R} ^{n}}$,

where the orthogonal group , consisting of reflections and rotations around the origin, is and is understood as the group of the displacements of the . ${\displaystyle O(n)}$ ${\displaystyle \mathbb {R} ^{n}}$ ${\displaystyle \mathbb {R} ^{n}}$

A rank crystallographic group ${\displaystyle n}$ is a discrete and co-compact subgroup

${\displaystyle \Gamma \subset Isom(\mathbb {R} ^{n})}$.

Co-compactness means that the group has a compact fundamental area.

sentences

1. If a crystallographic group is of rank , then the set of all displacements into a maximal Abelian subgroup is of finite index . ${\displaystyle \Gamma }$ ${\displaystyle n}$ ${\displaystyle \Gamma }$

2. There are only a finite number of isomorphism classes of crystallographic groups of rank . ${\displaystyle n}$

3. Two crystallographic groups are isomorphic if and only if they are conjugated within the group of affine transformations ; H. if there is one with . ${\displaystyle \Gamma _{1},\Gamma _{2}}$ ${\displaystyle A(n):=GL(n,\mathbb {R} )\ltimes \mathbb {R} ^{n}}$ ${\displaystyle A\in A(n)}$ ${\displaystyle A\Gamma _{1}A^{-1}=\Gamma _{2}}$

Zassenhaus's theorem

The 1st Bieberbach's theorem also has an inversion, with which crystallographic groups can be characterized abstractly within group theory. It was proven by Zassenhaus in 1947 .

Theorem: A group is a crystallographic group of rank if and only if it has a normal maximal Abelian subgroup of finite index. ${\displaystyle \Gamma }$ ${\displaystyle n}$ ${\displaystyle \mathbb {Z} ^{n}}$