- Klein four-group
In

mathematics , the**Klein four-group**(or just**Klein group**or**Vierergruppe**, often symbolized by the letter**V**) is the group Z_{2}× Z_{2}, thedirect product of two copies of thecyclic group of order 2 (or any isomorphic variant). It was named "Vierergruppe" byFelix 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: Z

_{4}(see also thelist 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:::

In 2D it is the

symmetry group of arhombus and of arectangle , 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 itspermutation representation on4 points::"V" = { identity, (1,2)(3,4), (1,3)(2,4), (1,4)(2,3) }

In this representation, V is a

normal subgroup of thealternating group "A"_{4}(and also thesymmetric group "S"_{4}) on 4 letters. In fact, it is the kernel of a surjective map from "S"_{4}to "S"_{3}.According toGalois 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 ofquartic equation s in terms of radicals.The Klein four-group as a subgroup of "A"

_{4}is not theautomorphism 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 thetopological ring ofsplit-complex number s.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 atNorthwestern University , best known for their song "Finite Simple Group (of Order Two)." [*[*]*http://youtube.com/watch?v=UTby_e4-Rhg YouTube - Finite Simple Group (of Order Two)*]**ee also***

Quaternion group

*Kleinian group **ources**

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