Definitions

Klein four-group

In mathematics, the Klein four-group (or just Klein group or Vierergruppe, often symbolized by the letter V) is the group Z2 × Z2, the direct product of two copies of the cyclic group of order 2 (or any isomorphic variant). It was named Vierergruppe by Felix Klein in his Vorlesungen über das Ikosaeder und die Auflösung der Gleichungen vom fünften Grade in 1884.

The Klein four-group is the smallest non-cyclic group. The only other group with four elements, up to isomorphism, is the cyclic group of order four: Z4 (see also the list of small groups).

All elements of the Klein group (except the identity) have order 2. It is abelian, and isomorphic to the dihedral group of order 4.

The Klein group's Cayley table is given by:

* 1 i j k
1 1 i j k
i i 1 k j
j j k 1 i
k k j i 1

In 2D it is the symmetry group of a rhombus and of a rectangle, the four elements being the identity, the vertical reflection, the horizontal reflection, and a 180 degree rotation.

In 3D there are three different symmetry groups which are algebraically the Klein four-group V:

• one with three perpendicular 2-fold rotation axes: D2
• one with a 2-fold rotation axis, and a perpendicular plane of reflection: C2h = D1d
• one with a 2-fold rotation axis in a plane of reflection (and hence also in a perpendicular plane of reflection): C2v = D1h

The three elements of order 2 in the Klein four-group are interchangeable: the automorphism group is the group of permutations of the three elements. This essential symmetry can also be seen by its permutation representation on 4 points:

V = { identity, (1,2)(3,4), (1,3)(2,4), (1,4)(2,3) }

In this representation, V is a normal subgroup of the alternating group A4 (and also the symmetric group S4) on 4 letters. In fact, it is the kernel of a surjective map from S4 to S3. According to Galois theory, the existence of the Klein four-group (and in particular, this representation of it) explains the existence of the formula for calculating the roots of quartic equations in terms of radicals.

The Klein four-group as a subgroup of A4 is not the automorphism group of any simple graph. It is, however, the automorphism group of a two-vertex graph where the vertices are connected to each other with two edges, making the graph non-simple. It is also the automorphism group of the following simple graph, but in the permutation representation { (1,2), (3,4), (1,2)(3,4) } where the points are labeled top-left, bottom-left, top-right, bottom-right:

The Klein four-group is the group of components of the group of units of the topological ring of split-complex numbers.

Another example of the Klein four-group is the multiplicative group { 1, 3, 5, 7 } with the action being multiplication modulo 8.

Field

The Klein four-group is isomorphic to the additive group of finite field GF(4):

` + | 0 1 A B       · | 0 1 A B`
` --+--------       --+--------`
` 0 | 0 1 A B       0 | 0 0 0 0`
` 1 | 1 0 B A       1 | 0 1 A B`
` A | A B 0 1       A | 0 A B 1`
` B | B A 1 0       B | 0 B 1 A`

In Popular Culture

The Klein Four is an a cappella singing group at Northwestern University, best known for their song "Finite Simple Group (of Order Two).