Klein four-group

In mathematics, the Klein four-group is an abelian group with four elements, in which each element is self-inverse (composing it with itself produces the identity) and in which composing any two of the three non-identity elements produces the third one. It can be described as the symmetry group of a non-square rectangle (with the three non-identity elements being horizontal reflection, vertical reflection and 180-degree rotation), as the group of bitwise exclusive-or operations on two-bit binary values, or more abstractly as $\mathbb {Z} _{2}\times \mathbb {Z} _{2}$ , the direct product of two copies of the cyclic group of order 2 by the Fundamental Theorem of Finitely Generated Abelian Groups. It was named Vierergruppe (German: [ˈfiːʁɐˌɡʁʊpə] ), meaning four-group) by Felix Klein in 1884.^[1] It is also called the Klein group, and is often symbolized by the letter $V$ or as $K_{4}$ .

"Vierergruppe" redirects here. For the four-person anti-Nazi Resistance group, see Vierergruppe (German Resistance).

The Klein four-group, with four elements, is the smallest group that is not cyclic. Up to isomorphism, there is only one other group of order four: the cyclic group of order 4. Both groups are abelian.

one with three perpendicular 2-fold rotation axes: the $D_{2}$

dihedral group

one with a 2-fold rotation axis, and a perpendicular plane of reflection: $C_{2\mathrm {h} }=D_{1\mathrm {d} }$

one with a 2-fold rotation axis in a plane of reflection (and hence also in a perpendicular plane of reflection): $C_{2\mathrm {v} }=D_{1\mathrm {h} }$ .

In two dimensions, the Klein four-group is the symmetry group of a rhombus and of rectangles that are not squares, the four elements being the identity, the vertical reflection, the horizontal reflection, and a 180° rotation.

In three dimensions, there are three different symmetry groups that are algebraically the Klein four-group:

Algebra[edit]

According to Galois theory, the existence of the Klein four-group (and in particular, the permutation 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.

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

If $\mathbb {R} ^{\times }$ denotes the multiplicative group of non-zero reals and $\mathbb {R} ^{+}$ the multiplicative group of positive reals, then $\mathbb {R} ^{\times }\times \mathbb {R} ^{\times }$ is the group of units of the ring $\mathbb {R} \times \mathbb {R}$ , and $\mathbb {R} ^{+}\times \mathbb {R} ^{+}$ is a subgroup of $\mathbb {R} ^{\times }\times \mathbb {R} ^{\times }$ (in fact it is the component of the identity of $\mathbb {R} ^{\times }\times \mathbb {R} ^{\times }$ ). The quotient group $(\mathbb {R} ^{\times }\times \mathbb {R} ^{\times })/(\mathbb {R} ^{+}\times \mathbb {R} ^{+})$ is isomorphic to the Klein four-group. In a similar fashion, the group of units of the split-complex number ring, when divided by its identity component, also results in the Klein four-group.