Klein four-group

In mathematics

Mathematics

Mathematics is the science and study of quantity, structure, space, and change. Mathematicians seek out patterns, formulate new conjectures, and establish truth by rigorous deduction from appropriately chosen axioms and definitions....

, the Klein four-group (or just Klein group or Vierergruppe, often symbolized by the letter V) is the group

Group (mathematics)

In mathematics, a group is an algebraic structure consisting of a set together with an operation that combines any two of its elements to form a third element. To qualify as a group, the set and the operation must satisfy a few conditions called group axioms, namely closure, associativity, identity...

  Z₂ × Z₂, the direct product

Direct product

In mathematics, one can often define a direct product of objectsalready known, giving a new one. This is generally the Cartesian product of the underlying sets, together with a suitably defined structure on the product set....

 of two copies of the cyclic group

Cyclic group

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 g such that, when written multiplicatively, every element of the group is a power of g .-Definition:A group G is called cyclic if there exists an element g...

 of order

Order (group theory)

In group theory, a branch of mathematics, the term order is used in two closely related senses:* the order of a group is its cardinality, i.e...

 2. It was named Vierergruppe by Felix Klein

Felix Klein

Felix Christian Klein was a German mathematician, known for his work in group theory, function theory, non-Euclidean geometry, and on the connections between geometry and group theory...

 in his Vorlesungen über das Ikosaeder und die Auflösung der Gleichungen vom fünften Grade in 1884.

The Klein four-group is the smallest non-cyclic group.

In mathematics

Mathematics

Mathematics is the science and study of quantity, structure, space, and change. Mathematicians seek out patterns, formulate new conjectures, and establish truth by rigorous deduction from appropriately chosen axioms and definitions....

, the Klein four-group (or just Klein group or Vierergruppe, often symbolized by the letter V) is the group

Group (mathematics)

In mathematics, a group is an algebraic structure consisting of a set together with an operation that combines any two of its elements to form a third element. To qualify as a group, the set and the operation must satisfy a few conditions called group axioms, namely closure, associativity, identity...

  Z₂ × Z₂, the direct product

Direct product

In mathematics, one can often define a direct product of objectsalready known, giving a new one. This is generally the Cartesian product of the underlying sets, together with a suitably defined structure on the product set....

 of two copies of the cyclic group

Cyclic group

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 g such that, when written multiplicatively, every element of the group is a power of g .-Definition:A group G is called cyclic if there exists an element g...

 of order

Order (group theory)

In group theory, a branch of mathematics, the term order is used in two closely related senses:* the order of a group is its cardinality, i.e...

 2. It was named Vierergruppe by Felix Klein

Felix Klein

Felix Christian Klein was a German mathematician, known for his work in group theory, function theory, non-Euclidean geometry, and on the connections between geometry and group theory...

 in his Vorlesungen über das Ikosaeder und die Auflösung der Gleichungen vom fünften Grade in 1884.

The Klein four-group is the smallest non-cyclic group. The only other group with four elements, up to

Up to

In mathematics, the phrase "up to xxxx" indicates that members of an equivalence class are to be regarded as a single entity for some purpose. "xxxx" describes a property or process which transforms an element into one from the same equivalence class, i.e. one to which it is considered equivalent...

 isomorphism, is Z₄, the cyclic group of order four (see also the list of small groups).

All non-identity

Identity element

In mathematics, an identity element is a special type of element of a set with respect to a binary operation on that set. It leaves other elements unchanged when combined with them...

 elements of the Klein group have order 2.
It is abelian

Abelian group

An abelian group, also called a commutative group, is a group in which the result of applying the group operation to two group elements does not depend on their order . Abelian groups generalize the arithmetic of addition of integers...

, and isomorphic to the dihedral group

Dihedral group

In mathematics, a dihedral group is the group of symmetries of a regular polygon, including both rotations and reflections. Dihedral groups are among the simplest examples of finite groups, and they play an important role in group theory, geometry, and chemistry.See also: Dihedral symmetry in three...

 of order 2.

The Klein group's Cayley table

Cayley table

A Cayley table, after the 19th century British mathematician Arthur Cayley, describes the structure of a finite group by arranging all the possible products of all the group's elements in a square table reminiscent of an addition or multiplication table...

 is given by:

{| border="1" cellpadding="11" style="border-collapse: collapse; border: 1px #aaa solid;"

!style="background:#efefef;"| *
!style="background:#efefef;"| 1
!style="background:#efefef;"| i
!style="background:#efefef;"| j
!style="background:#efefef;"| k
|-
!style="background:#efefef;"| 1
| 1 || i || j || k
|-
!style="background:#efefef;"| i
| i || 1 || k || j
|-
!style="background:#efefef;"| j
| j || k || 1 || i
|-
!style="background:#efefef;"| k
| k || j || i || 1
|}

In 2D it is the symmetry group

Symmetry group

The symmetry group of an object is the group of all isometries under which it is invariant with composition as the operation...

 of a rhombus

Rhombus

In geometry, a rhombus or rhomb is a quadrilateral whose four sides all have the same length. The rhombus is often called a diamond, after the diamonds suit in playing cards, or a lozenge, though the latter sometimes refers specifically to a rhombus with a 45° angle.In general, a polygon whose...

 and of a rectangle

Rectangle

In Euclidean geometry, the term rectangle normally refers to a quadrilateral with four right angles. This is a simple rectangle. A simple rectangle with vertices ABCD would be denoted as ....

, the four elements being the identity, the vertical reflection, the horizontal reflection, and a 180 degree rotation.

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:

V = { identity, (1,2)(3,4), (1,3)(2,4), (1,4)(2,3) }

In this representation, V is a normal subgroup

Normal subgroup

In mathematics, more specifically in abstract algebra, a normal subgroup is a special kind of subgroup. Normal subgroups are important because they can be used to construct quotient groups from a given group....

 of the alternating group

Alternating group

In mathematics, an alternating group is the group of even permutations of a finite set. The alternating group on the set {1,...,n} is called the alternating group of degree n, or the alternating group on n letters and denoted by A_n or Alt.For instance, the alternating group of degree 4...

 A₄
(and also the symmetric group

Symmetric group

In mathematics, the symmetric group on a set is the group consisting of all automorphisms of the set with function composition as the group operation....

 S₄) on 4 letters. In fact, it is the kernel of a surjective map from S₄ to S₃.
According to Galois theory

Galois theory

In mathematics, more specifically in abstract algebra, Galois theory, named after Évariste Galois, provides a connection between field theory and group theory...

, the existence of the Klein four-group (and in particular, this representation of it) explains the existence of the formula for calculating the roots of quartic equations in terms of radical

Radical of an algebraic group

In mathematics, an algebraic group G contains a unique maximal normal solvable subgroup; and this subgroup is closed. Its identity component is called the radical of G.- External links :*, Encyclopaedia of Mathematics...

s, as established by Lodovico Ferrari

Lodovico Ferrari

Lodovico Ferrari was an Italian mathematician.Born in Milan, Italy, grandfather, Bartholomew Ferrari was forced out of Milan to bologna, He settled in Bologna, Italy and he began his career as the servant of Gerolamo Cardano. He was extremely bright, so Cardano started teaching him mathematics...

:
the map corresponds to the resolvent cubic, in terms of Lagrange resolvents.

The Klein four-group as a subgroup of A₄ is not the automorphism group of any simple graph

Graph theory

In mathematics and computer science, graph theory is the study of graphs: mathematical structures used to model pairwise relations between objects from a certain collection. A "graph" in this context refers to a collection of vertices or 'nodes' and a collection of edges that connect pairs of...

. It is, however, the automorphism group of a two-vertex graph where the vertices are connected to each other with two edges, making the graph non-simple. It is also the automorphism group of the following simple graph, but in the permutation representation { , (1,2), (3,4), (1,2)(3,4) } where the points are labeled top-left, bottom-left, top-right, bottom-right:

The Klein four-group is the group of components

Identity component

In mathematics, the identity component of a topological group G is the connected component G₀ that contains the identity element e....

 of the group of units of the topological ring

Topological ring

In mathematics, a topological ring is a ring R which is also a topological space such that both the addition and the multiplication are continuous as mapswhere R × R carries the product topology.- General comments :...

 of split-complex number

Split-complex number

In abstract algebra, the split-complex numbers are a two-dimensional commutative algebra over the real numbers different from the complex numbers. Every split-complex number has the formwhere x and y are real numbers...

s.

Another example of the Klein four-group is the multiplicative group

Multiplicative group

In mathematics and group theory the term multiplicative group refers to one of the following concepts, depending on the context*any group whose binary operation is written in multiplicative notation ,*the underlying group under multiplication of the invertible elements of a field, ring, or other...

 { 1, 3, 5, 7 } with the action being multiplication modulo 8

Modular arithmetic

In mathematics, modular arithmetic is a system of arithmetic for integers, where numbers "wrap around" after they reach a certain value—the modulus...

.

In the construction of finite ring

Finite ring

In mathematics, more specifically abstract algebra, a finite ring is a ring that has a finite number of elements....

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