In mathematics, the Klein four-group (or just Klein group or Vierergruppe, often symbolized by the letter V) is the group Z₂ × Z₂, the direct product of two copies of the cyclic group of order 2 (or any isomorphic variant). It was named Vierergruppe by Felix Klein in his Vorlesungen über das Ikosaeder und die Auflösung der Gleichungen vom fünften Grade in 1884. Euclid, detail from The School of Athens by Raphael. ... In mathematics, a group is a set, together with a binary operation, such as multiplication or addition, satisfying certain axioms, detailed below. ... In mathematics, one can often define a direct product of objects already known, giving a new one. ... In group theory, a cyclic group is a group that can be generated by a single element, in the sense that the group has an element a (called a generator of the group) such that, when written multiplicatively, every element of the group is a power of a (or na... In group theory, the term order is used in two closely related senses: the order of a group is its cardinality, i. ... In mathematics, an isomorphism (in Greek isos = equal and morphe = shape) is a kind of mapping between objects, devised by Eilhard Mitscherlich, which shows a relation between two properties or operations. ... Felix Christian Klein (April 25, 1849 – June 22, 1925) was a German mathematician. ...

The Klein four-group is the smallest non-cyclic group. The only other group with four elements, up to isomorphism, is the cyclic group of order four: Z₄ (see also the list of small groups). The following list in mathematics contains the finite groups of small order up to group isomorphism. ...

All elements of the Klein group (except the identity) have order 2. It is abelian, and isomorphic to the dihedral group of order 4. In mathematics, an identity element (or neutral element) is a special type of element of a set with respect to a binary operation on that set. ... In mathematics, an abelian group, also called a commutative group, is a group such that for all a and b in G. In other words, the order of elements in a product doesnt matter. ... This article may be confusing for some readers, and should be edited to enhance clarity. ...

The Klein group's multiplication table is given by: A Cayley table is a representation of a product defined on a set G. It is a group-theoretic generalization of an addition or a multiplication table. ...

In 2D it is the symmetry group of a rhombus and of a rectangle, the four elements being the identity, the vertical reflection, the horizontal reflection, and a 180 degree rotation. The symmetry group of an object (e. ... This shape is a rhombus In geometry, a rhombus (also known as a rhomb) is a quadrilateral in which all of the sides are of equal length. ... In geometry, a rectangle is defined as a quadrilateral polygon in which all four angles are right angles. ...

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: In mathematics, an automorphism is an isomorphism from a mathematical object to itself. ... In mathematics, groups are often used to describe symmetries of objects. ...

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. According to Galois theory, the existence of the Klein four-group (and in particular, this particular representation) explains the existence of the formula for calculating the roots of quartic equations in terms of radicals. In mathematics, a normal subgroup N of a group G is a subgroup invariant under conjugation; that is, for each element n in N and each g in G, the element g−1ng is still in N. The statement N is a normal subgroup of G is written: . There are... In mathematics an alternating group is the group of even permutations of a finite set. ... In mathematics, the symmetric group on a set X, denoted by SX or Sym(X), is the group whose underlying set is the set of all bijective functions from X to X, in which the group operation is that of composition of functions, i. ... In mathematics, Galois theory is a branch of abstract algebra. ... In mathematics, a quartic equation is the result of setting a quartic function equal to zero. ...

One can also think of the Klein four-group as the automorphism group of the following graph: In mathematics, an automorphism is an isomorphism from a mathematical object to itself. ...

The Klein four-group is the discrete part {1, j, −1, −j} of the group of units of the split-complex number ring. Image File history File links Klein_4-Group_Graph. ... In mathematics, the split-complex numbers are an extension of the real numbers defined analogously to the complex numbers. ...

Compare:
quaternion group
Kleinian group. Cycle diagram of Q. Each color specifies a series of powers of any element connected to the identity element (1). ... In mathematics, a Kleinian group is a finitely generated discrete group Γ of conformal (i. ...