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, 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 (, meaning four-group) by Felix Klein in 1884. It is also called the Klein group.

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.

Presentations

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. It is, however, an abelian group, and isomorphic to the dihedral group, using the geometric convention); other than the group of order 2, it is the only dihedral group that is abelian.

The Klein four-group is also isomorphic to the direct sum, so that it can be represented as the pairs under component-wise addition modulo 2 (or equivalently the bit strings 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; the empty set is the group's identity element in this case.

Another numerical construction of the Klein four-group is the set , with the operation being multiplication modulo 8. Here a is 3, b is 5, and is .

The Klein four-group also has a representation as real matrices with the operation being matrix multiplication:

On a Rubik's Cube represents the group of Slice layer states in which two sets of two edges are swapped.

Geometry

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:

• one with three perpendicular 2-fold rotation axes: the dihedral group

• .

Permutation representation

The three elements of order two in the Klein four-group are interchangeable: the automorphism group of V is thus the group of permutations of these three elements, that is, the symmetric group.

The Klein four-group's permutations of its own elements can be thought of abstractly as its permutation representation on four points:

1. {(), (1,2)(3,4), (1,3)(2,4), (1,4)(2,3)}

is a normal subgroup of the alternating group (and also the symmetric groupthat appears as a Galois group. In fact, it is the kernel of a surjective group homomorphism.

Other representations within S4 are:

They are not normal subgroups of S4.

Algebra

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 radicalss, as established by Lodovico Ferrari 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.

the multiplicative group of positive reals is the group of units (in fact it is the component of the identity). The quotient group 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.

Graph theory

Among the simple connected graphs, the one with the fewest entities 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 with even 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.

Music

In music composition, the four-group is the basic group of permutations in the twelve-tone technique. In that instance, the Cayley table is written

SATOR square

The SATOR square is a 5×5 word square :It satisfies the following constraints :

forms a Klein group.

See also

• Quaternion group

• List of small groups

References

Further reading

• M. A. Armstrong (1988) Groups and Symmetry, Springer Verlag, [ page 53].

• W. E. Barnes (1963) Introduction to Abstract Algebra, D.C. Heath & Co., page 20.

External links