Klein Four-group

In mathematics, the Klein four-group (or just Klein group or Vierergruppe (English: four-group), 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 Z'₄, 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 but this splitting is not natural – it requires choosing two of the (non-zero) elements, in order, as the generators, and different choices give different isomorphisms.

The Klein group's Cayley table is given by:

*	1	a	b	ab
1	1	a	b	ab
a	a	1	ab	b
b	b	ab	1	a
ab	ab	b	a	1

An elementary construction of the Klein four-group is the multiplicative group { 1, 3, 5, 7 } with the action being multiplication modulo 8. Here a is 3, b is 5, and ab is 3 ×5 = 15 ≡ 7 (mod 8).

In 2D it is the symmetry group of a rhombus and of a rectangle which are not squares, 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: 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 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 corresponds to the resolvent cubic, in terms of Lagrange resolvents.

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:

In the construction of finite rings, eight of the eleven rings with four elements have the Klein four-group as their additive substructure.

The basic group of permutations in the twelve-tone technique is a four-group (Babbitt 1960, 253):

In interval analysis the set of intervals on the real line is identified with the points (a,b) of the Cartesian plane with Euclidean distance. The Klein four-group arises from the multiplicative structure on the direct sum . The x-axis and the y-axis are ideals of the ring of intervals where is the identity so quadrant I is the component of the identity. The Klein four-group is the quotient of the multiplicative group of intervals that do not end on an axis by the subgroup consisting of intervals represented by points in quadrant I. Similarly, the ring of intervals is normalized to the split-complex number ring which also produces the Klein four-group as the quotient of its group of units by its identity component.