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). Since in German the cardinal "four" starts with the letter V (vier) Klein introduced the symbol V.

Multiplication table
The multiplication table of the group is


 * {| class="wikitable"

! V ! e ! a ! b ! c ! e ! a ! b ! c
 * style="background:#99FFCC;width:24px;height:24px" align="center"|e || style="background:yellow;width:24px;height:24px" align="center"|a || style="background:yellow;width:24px;height:24px" align="center"|b || style="background:yellow;width:24px;height:24px" align="center"|c
 * style="background:yellow;width:24px;height:24px" align="center"|a || style="background:#99FFCC;width:24px;height:24px" align="center"|e || style="background:yellow;width:24px;height:24px" align="center"|c || style="background:yellow;width:24px;height:24px" align="center"|b
 * style="background:yellow;width:24px;height:24px" align="center"|b || style="background:yellow;width:24px;height:24px" align="center"|c || style="background:#99FFCC;width:24px;height:24px" align="center"|e || style="background:yellow;width:24px;height:24px" align="center"|a
 * style="background:yellow;width:24px;height:24px" align="center"|c || style="background:yellow;width:24px;height:24px" align="center"|b || style="background:yellow;width:24px;height:24px" align="center"|a || style="background:#99FFCC;width:24px;height:24px" align="center"|e
 * }

This table is symmetric, meaning that the elements commute: ab = ba, etc. The elements not equal to the identity e have the property g2 = e (these elements are of order 2).

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, &minus;y, &minus;z)
 * b:    (&minus;x, y, &minus;z)
 * c:    (&minus;x, &minus;y, z)