The Klein four-group is also defined by the group presentation All non-identity elements of the Klein group have order 2, thus 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 of order 4, i.e. D4 ; 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 ; with 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, i.e. over a field of sets with four elements, e.g. ; 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, bis 5, and is. The Klein four-group has a representation as 2x2 real matrices with the operation being matrix multiplication:
Geometry
Geometrically, 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 degree rotation. In three dimensions there are three different symmetry groups that 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:
one with a 2-fold rotation axis in a plane of reflection :.
Permutation representation
The three elements of order two in the Klein four-group are interchangeable: the automorphism group of V is the group of permutations of these three elements. 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, V is a normal subgroup of the alternating group A4 on four letters. In fact, it is the kernel of a surjective group homomorphism from S4 to S3. Other representations within S4 are: They are not normal subgroups of S4.
Algebra
According to Galois theory, the existence of the Klein four-group 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. In the construction of finite rings, eight of the eleven rings with four elements have the Klein four-group as their additive substructure. If R× denotes the multiplicative group of non-zero reals and R+ the multiplicative group of positive reals, R× × R× is the group of units of the ring, and is a subgroup of . 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
The simplest simpleconnectedgraph 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.