Klein four-group

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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. 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 Z4, the cyclic group of order four (see also the list of small groups).

All non-identity elements of the Klein group have order 2. It is abelian, and isomorphic to the dihedral group of order (cardinality) 4. It is also isomorphic to the direct sum :\mathbb{Z}_2 \oplus \mathbb{Z}_2

The Klein group's Cayley table is given by:

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:

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, as established by Lodovico Ferrari: the map S_4 \to S_3 corresponds to the resolvent cubic, in terms of Lagrange resolvents.

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