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: ''D''₂

★ one with a 2-fold rotation axis, and a perpendicular plane of reflection: ''C''_2h = ''D''_1d

★ one with a 2-fold rotation axis in a plane of reflection (and hence also in a perpendicular plane of reflection): ''C''_2v = ''D''_1h
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 ''A''₄
(and also the symmetric group ''S''₄) on 4 letters. In fact, it is the kernel of a surjective map from ''S''₄ to ''S''₃.
According to Galois theory, the existence of the Klein four-group
(and in particular, this particular representation)
explains the existence of the formula for calculating the roots of
quartic equations in terms of radicals.
One can also think of the Klein four-group as the automorphism group of the following graph:
::
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.

Contents

Field

See also

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

See also

★ Dihedral group

★ Quaternion group

★ Kleinian group