Klein four-group
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
, 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
The Klein four-group, with four elements, is the smallest group that is not cyclic.
The Klein group's Cayley table is given by: The Klein four-group is also defined by the group presentation All non-identity elements of the Klein group have order 2, so any two non-identity elements can serve as generators in the above presentation.
The Klein four-group is the smallest non-cyclic group.
The Klein four-group is also isomorphic to the direct sum
, so that it can be represented as the pairs {(0,0), (0,1), (1,0), (1,1)} under component-wise addition modulo 2 (or equivalently the bit strings {00, 01, 10, 11} under bitwise XOR), with (0,0) being the group's identity element.
The Klein four-group is thus an example of an elementary abelian 2-group, which is also called a Boolean group.
The Klein four-group is thus also the group generated by the symmetric difference as the binary operation on the subsets of a powerset of a set with two elements—that is, over a field of sets with four elements, such as
; the empty set is the group's identity element in this case.
Another numerical construction of the Klein four-group is the set { 1, 3, 5, 7 }, with the operation being multiplication modulo 8.
The Klein four-group also has a representation as 2 × 2 real matrices with the operation being matrix multiplication: On a Rubik's Cube, the "4 dots" pattern can be made in three ways (for example, M2 U2 M2 U2 F2 M2 F2), depending on the pair of faces that are left blank; these three positions together with the solved position form an example of the Klein group, with the solved position serving as the identity.
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.
The Klein four-group's permutations of its own elements can be thought of abstractly as its permutation representation on four points: In this representation,
In fact, it is the kernel of a surjective group homomorphism from
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
denotes the multiplicative group of non-zero reals and
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.
Among the simple connected graphs, the simplest (in the sense of having the fewest entities) that admits the Klein four-group as its automorphism group is the diamond graph shown below.
It is also the automorphism group of some other graphs that are simpler in the sense of having fewer entities.
These include the graph with four vertices and one edge, which remains simple but loses connectivity, and the graph with two vertices connected to each other by two edges, which remains connected but loses simplicity.
In music composition, the four-group is the basic group of permutations in the twelve-tone technique.
