# Vierergruppe

In group theory, a branch of mathematics, the **Vierergruppe** (German, meaning group of four) is the smallest non-cyclic group. It is an Abelian (commutative) group of order 4.

The group was given his name by Felix Klein in his 1884 lectures "*Vorlesungen über das Ikosaeder und die Auflösung der Gleichungen vom fünften Grade*" (lectures on the Icosahedron and the solution of equations of the fifth degree).^{[1]} Since in German the cardinal "four" starts with the letter V (vier) Klein introduced the symbol *V*.

## [edit] Multiplication table

The multiplication table of the group is

*V**e**a**b**c**e**e**a**b**c**a**a**e**c**b**b**b**c**e**a**c**c**b**a**e*

This table is symmetric, meaning that the elements commute: *ab* = *ba*, etc. The elements not equal to the identity *e* have the property *g*^{2} = *e* (these elements are of order 2).

## [edit] Example

The classic example of a Vierergruppe, first given by Klein, is the set of rotations over 180° around three orthogonal axes, for instance Cartesian axes, mapping (*x*, *y*, *z*) to

*e:*(*x*,*y*,*z*),*a:*(*x*,*−y*,*−z*)*b:*(*−x*,*y*,*−z*)*c:*(*−x*,*−y*,*z*)