Finite group

A finite group occurs in the mathematical discipline of group theory. Finite groups are groups of the set M contains a finite number of elements.

Axioms

The assumption of finiteness allows a simplified system of axioms (see van der Waerden pp. 15-17 ):

A couple with a finite set and an inner two -digit shortcut called group if the following axioms are satisfied:

• Associativity: For all group elements, and the following applies:

• Uniqueness of the reduction: Off, as well as the following:.

From the uniqueness of the reduction follows that the left - and right multiplication and injective, which implies the surjectivity because of the finiteness. Therefore, it is with one, resulting in the existence of a neutral element, and then with, indicating the existence of the inverse elements.

Simple groups

Every finite group is composed of a finite number of finite simple groups. However, this composition can be complicated, and despite the knowledge of the building blocks ( the simple groups ) we are still far from knowing all finite groups.

Since 1982, the finite simple groups are completely classified:

• Almost all of these groups can be one of 18 families of finite simple groups.
• There are 26 exceptions - these groups are referred to as sporadic groups.

Examples

Finite groups are given by the cyclic groups or permutation groups (see: Symmetric group, alternating group).

Among the sporadic groups, the Conway group, the baby monster and the monster group includes ( with almost 1054 members, the largest sporadic group).

Applications

Symmetries of bodies, particularly in molecular physics are described by point groups; Symmetries of crystals by 230 different space groups.