Klein four-group

# Klein four-group

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In mathematics
Mathematics
Mathematics is the study of quantity, space, structure, and change. Mathematicians seek out patterns and formulate new conjectures. Mathematicians resolve the truth or falsity of conjectures by mathematical proofs, which are arguments sufficient to convince other mathematicians of their validity...

, 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 of groups
In the mathematical field of group theory, the direct product is an operation that takes two groups and and constructs a new group, usually denoted...

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., the number of its elements....

2. It was named Vierergruppe by Felix Klein
Felix Klein
Christian Felix 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 x" means "disregarding a possible difference in  x".For instance, when calculating an indefinite integral, one could say that the solution is f "up to addition by a constant," meaning it differs from f, if at all, only by some constant.It indicates that...

isomorphism
Isomorphism
In abstract algebra, an isomorphism is a mapping between objects that shows a relationship between two properties or operations.  If there exists an isomorphism between two structures, the two structures are said to be isomorphic.  In a certain sense, isomorphic structures are...

, 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
In abstract algebra, 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 (cardinality) 4. It is also isomorphic to the direct sum
Direct sum
In mathematics, one can often define a direct sum of objectsalready known, giving a new one. This is generally the Cartesian product of the underlying sets , together with a suitably defined structure. More abstractly, the direct sum is often, but not always, the coproduct in the category in question...

:

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;"| a
!style="background:#efefef;"| b
!style="background:#efefef;"| ab
|-
!style="background:#efefef;"| 1
| 1 || a || b || ab
|-
!style="background:#efefef;"| a
| a || 1 || ab || b
|-
!style="background:#efefef;"| b
| b || ab || 1 || a
|-
!style="background:#efefef;"| ab
| ab || b || a || 1
|}

An elementary construction 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 \scriptstyle\mathfrak \,\! whose binary operation is written in multiplicative notation ,*the underlying group under multiplication of the invertible elements of...

{ 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....

. Here a is 3, b is 5, and ab is 3×5=15≡7 (mod 8).

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 Euclidean geometry, a rhombus or rhomb is a convex 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.Every...

and of a rectangle
Rectangle
In Euclidean plane geometry, a rectangle is any quadrilateral with four right angles. The term "oblong" is occasionally used to refer to a non-square rectangle...

which are not squares
Square (geometry)
In geometry, a square is a regular quadrilateral. This means that it has four equal sides and four equal angles...

, 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: C2h = D1d
• one with a 2-fold rotation axis in a plane of reflection (and hence also in a perpendicular plane of reflection): C2v = D1h

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 abstract algebra, a normal subgroup is a subgroup which is invariant under conjugation by members of the group. Normal subgroups can be used to construct quotient groups from a given group....

of the alternating group A4
(and also the symmetric group
Symmetric group
In mathematics, the symmetric group Sn on a finite set of n symbols is the group whose elements are all the permutations of the n symbols, and whose group operation is the composition of such permutations, which are treated as bijective functions from the set of symbols to itself...

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
The radical of an algebraic group is the identity component of its maximal normal solvable subgroup.- 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 (mathematics)
In mathematics, a graph is an abstract representation of a set of objects where some pairs of the objects are connected by links. The interconnected objects are represented by mathematical abstractions called vertices, and the links that connect some pairs of vertices are called edges...

. 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 of G that contains the identity element of the group...

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.

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.

The basic group of permutations in the twelve-tone technique
Twelve-tone technique
Twelve-tone technique is a method of musical composition devised by Arnold Schoenberg...

is a four-group (Babbitt 1960, 253):
 S I: R: RI: I: S RI R R: RI S I RI: R I S