Klein four-group

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For the a cappella group, see The Klein Four.

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 multiplication 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. 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:

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The Klein four-group is the discrete part {1, j, −1, −j} of the group of units of the split-complex number ring.

Compare:
quaternion group
Kleinian group.