In _{4}.
The Klein four-group, with four elements, is the smallest group that is not a

_{4} (or D_{2}, 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

_{2}
*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 (and hence also in a perpendicular plane of reflection): .

_{4}
(and also the _{4}) on four letters. In fact, it is the _{4} to S_{3}.
Other representations within S_{4} are:
:
:
:
They are not normal subgroups of S_{4.}

^{×} denotes the multiplicative group of non-zero reals and R^{+} the multiplicative group of positive reals, R^{×} × R^{×} is the

mathematics
Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities and their changes (cal ...

, the Klein four-group is a group
A group is a number
A number is a mathematical object used to counting, count, measurement, measure, and nominal number, label. The original examples are the natural numbers 1, 2, 3, 4, and so forth. Numbers can be represented in language with ...

with four elements, in which each element is self-inverse
Image:Involution.svg, An involution is a function f:X\to X that, when applied twice, brings one back to the starting point.
In mathematics, an involution, or an involutory function, is a function (mathematics), function that is its own inverse fun ...

(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
In group theory
The popular puzzle Rubik's cube invented in 1974 by Ernő Rubik has been used as an illustration of permutation group">Ernő_Rubik.html" ;"title="Rubik's cube invented in 1974 by Ernő Rubik">Rubik's cube invented in 1974 ...

of a non-square rectangle
In Euclidean geometry, Euclidean plane geometry, a rectangle is a quadrilateral with four right angles. It can also be defined as: an equiangular quadrilateral, since equiangular means that all of its angles are equal (360°/4 = 90°); or a para ...

(with the three non-identity elements being horizontal and vertical reflection and 180-degree rotation),
as the group of bitwise exclusive or
Exclusive or or exclusive disjunction is a Logical connective, logical operation that is true if and only if its arguments differ (one is true, the other is false).
It is Table of logic symbols, symbolized by the prefix operator J and by the ...

operations on two-bit binary values,
or more abstractly as , the direct productIn mathematics
Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). It ha ...

of two copies of the cyclic group
In group theory
The popular puzzle Rubik's cube invented in 1974 by Ernő Rubik has been used as an illustration of permutation group">Ernő_Rubik.html" ;"title="Rubik's cube invented in 1974 by Ernő Rubik">Rubik's cube invented in 1974 by Er ...

of order
Order, ORDER or Orders may refer to:
* Orderliness
Orderliness is a quality that is characterized by a person’s interest in keeping their surroundings and themselves well organized, and is associated with other qualities such as cleanliness a ...

2.
It was named ''Vierergruppe'' (meaning four-group) by Felix Klein
Christian Felix Klein (; 25 April 1849 – 22 June 1925) was a German mathematician and mathematics educator, known for his work with group theory, complex analysis, non-Euclidean geometry, and on the associations between geometry and group ...

in 1884.
It is also called the Klein group, and is often symbolized by the letter V or as Kcyclic group
In group theory
The popular puzzle Rubik's cube invented in 1974 by Ernő Rubik has been used as an illustration of permutation group">Ernő_Rubik.html" ;"title="Rubik's cube invented in 1974 by Ernő Rubik">Rubik's cube invented in 1974 by Er ...

. There is only one other group of order four, up to isomorphism
In mathematics
Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities and ...

, the cyclic group of order 4. Both are abelian group
In mathematics
Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities an ...

s. The smallest non-abelian group is the symmetric group of degree 3, which has order 6.
Presentations

The Klein group'sCayley table Named after the 19th century British
British may refer to:
Peoples, culture, and language
* British people, nationals or natives of the United Kingdom, British Overseas Territories, and Crown Dependencies.
** Britishness, the British identity a ...

is given by:
The Klein four-group is also defined by the group presentation
In mathematics, a presentation is one method of specifying a group (mathematics), group. A presentation of a group ''G'' comprises a set ''S'' of generating set of a group, generators—so that every element of the group can be written as a produ ...

:$\backslash mathrm\; =\; \backslash left\backslash langle\; a,b\; \backslash mid\; a^2\; =\; b^2\; =\; (ab)^2\; =\; e\; \backslash right\backslash rangle.$
All non-identity
Identity may refer to:
Social sciences
* Identity (social science), personhood or group affiliation in psychology and sociology
Group expression and affiliation
* Cultural identity, a person's self-affiliation (or categorization by others ...

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
In group theory
The popular puzzle Rubik's cube invented in 1974 by Ernő Rubik has been used as an illustration of permutation group">Ernő_Rubik.html" ;"title="Rubik's cube invented in 1974 by Ernő Rubik">Rubik's cube invented in 1974 by Er ...

. It is however an abelian group
In mathematics
Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities an ...

, and isomorphic to the dihedral group
In mathematics, a dihedral group is the group (mathematics), group of symmetry, symmetries of a regular polygon, which includes rotational symmetry, rotations and reflection symmetry, reflections. Dihedral groups are among the simplest example ...

of order (cardinality) 4, i.e. Ddirect sum
The direct sum is an operation from abstract algebra
In algebra, which is a broad division of mathematics, abstract algebra (occasionally called modern algebra) is the study of algebraic structures. Algebraic structures include group (mathema ...

, so that it can be represented as the pairs under component-wise addition modulo 2 (or equivalently the bit strings under bitwise XOR
In computer programming, a bitwise operation operates on a bit string, a bit array or a Binary numeral system, binary numeral (considered as a bit string) at the level of its individual bits. It is a fast and simple action, basic to the higher le ...

); 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
In mathematics, specifically in group theory, an elementary abelian group (or elementary abelian ''p''-group) is an abelian group in which every nontrivial element has order ''p''. The number ''p'' must be prime, and the elementary abelian groups ...

. The Klein four-group is thus also the group generated by the symmetric difference
In mathematics
Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). It ...

as the binary operation on the subset
In mathematics
Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities a ...

s of a powerset
Image:Hasse diagram of powerset of 3.svg, 250px, The elements of the power set of order theory, ordered with respect to Inclusion (set theory), inclusion.
In mathematics, the power set (or powerset) of a Set (mathematics), set is the set of al ...

of a set with two elements, i.e. over a field of sets
In mathematics
Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities and ...

with four elements, e.g. $\backslash $; the empty set #REDIRECT Empty set #REDIRECT Empty set#REDIRECT Empty set
In mathematics
Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry ...

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 has a representation as 2×2 real matrices with the operation being matrix multiplication:
:$e\; =\backslash begin\; 1\; \&\; 0\backslash \backslash \; 0\; \&\; 1\; \backslash end,\backslash quad\; a\; =\; \backslash begin\; 1\; \&\; 0\backslash \backslash \; 0\; \&\; -1\; \backslash end,\backslash quad\; b\; =\; \backslash begin\; -1\; \&\; 0\backslash \backslash \; 0\; \&\; 1\; \backslash end,\backslash quad\; c\; =\; \backslash begin\; -1\; \&\; 0\backslash \backslash \; 0\; \&\; -1\; \backslash end$
Geometry

Geometrically, in two dimensions the Klein four-group is thesymmetry group
In group theory
The popular puzzle Rubik's cube invented in 1974 by Ernő Rubik has been used as an illustration of permutation group">Ernő_Rubik.html" ;"title="Rubik's cube invented in 1974 by Ernő Rubik">Rubik's cube invented in 1974 ...

of a rhombus
In plane Euclidean geometry
Euclidean geometry is a mathematical system attributed to Alexandrian Greek mathematics , Greek mathematician Euclid, which he described in his textbook on geometry: the ''Euclid's Elements, Elements''. Euclid's ...

and of rectangle
In Euclidean geometry, Euclidean plane geometry, a rectangle is a quadrilateral with four right angles. It can also be defined as: an equiangular quadrilateral, since equiangular means that all of its angles are equal (360°/4 = 90°); or a para ...

s that are not squares
In geometry, a square is a regular polygon, regular quadrilateral, which means that it has four equal sides and four equal angles (90-degree (angle), degree angles, or 100-gradian angles or right angles). It can also be defined as a rectangle in ...

, 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: DPermutation representation

The three elements of order two in the Klein four-group are interchangeable: theautomorphism group
In mathematics
Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities and th ...

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
In mathematics
Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). It h ...

on four points:
: V =
In this representation, V is a normal subgroup
In abstract algebra
In algebra, which is a broad division of mathematics, abstract algebra (occasionally called modern algebra) is the study of algebraic structures. Algebraic structures include group (mathematics), groups, ring (mathematics), ...

of the alternating group
In mathematics
Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities and t ...

Asymmetric group
In abstract algebra
In algebra, which is a broad division of mathematics, abstract algebra (occasionally called modern algebra) is the study of algebraic structures. Algebraic structures include group (mathematics), groups, ring (mathemati ...

Skernel
Kernel may refer to:
Computing
* Kernel (operating system), the central component of most operating systems
* Kernel (image processing), a matrix used for image convolution
* Compute kernel, in GPGPU programming
* Kernel method, in machine learnin ...

of a surjective group homomorphism
In mathematics
Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities and ...

from SAlgebra

According toGalois theory
In mathematics, Galois theory, originally introduced by Évariste Galois, provides a connection between field (mathematics), field theory and group theory. This connection, the fundamental theorem of Galois theory, allows reducing certain problems ...

, 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 equation
In mathematics
Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). It h ...

s in terms of radical
Radical may refer to:
Arts and entertainment Music
*Radical (mixtape), ''Radical'' (mixtape), by Odd Future, 2010
*Radical (Smack album), ''Radical'' (Smack album), 1988
*"Radicals", a song by Tyler, The Creator from the 2011 album ''Goblin (album ...

s, as established by Lodovico Ferrari
Lodovico de Ferrari (2 February 1522 – 5 October 1565) was an Italian
Italian may refer to:
* Anything of, from, or related to the country and nation of Italy
** Italians, an ethnic group or simply a citizen of the Italian Republic
** Ital ...

:
the map corresponds to the resolvent cubic, in terms of Lagrange resolvents
In Galois theory
In mathematics
Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analys ...

.
In the construction of finite ringIn mathematics
Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). It ha ...

s, eight of the eleven rings with four elements have the Klein four-group as their additive substructure.
If Rgroup of units
In the branch of abstract algebra known as ring theory
In algebra, ring theory is the study of ring (mathematics), rings—algebraic structures in which addition and multiplication are defined and have similar properties to those operations def ...

of the ring , and is a subgroup of (in fact it is the component of the identity of ). The quotient group
A quotient group or factor group is a mathematical group (mathematics), group obtained by aggregating similar elements of a larger group using an equivalence relation that preserves some of the group structure (the rest of the structure is "factore ...

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 simplestsimple
Simple or SIMPLE may refer to:
* Simplicity, the state or quality of being simple
Arts and entertainment
* ''Simple'' (album), by Andy Yorke, 2008, and its title track
* "Simple" (Florida Georgia Line song), 2018
* "Simple", a song by Johnn ...

connected graph
In mathematics
Mathematics (from Greek: ) includes the study of such topics as numbers ( and ), formulas and related structures (), shapes and spaces in which they are contained (), and quantities and their changes ( and ). There is no gen ...

that admits the Klein four-group as its automorphism group
In mathematics
Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities and th ...

is the diamond graph
Diamond is a Allotropes of carbon, solid form of the element carbon with its atoms arranged in a crystal structure called diamond cubic. At Standard conditions for temperature and pressure, room temperature and pressure, another solid form of ...

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.
Music

Inmusic composition
Musical composition can refer to an Originality, original piece or work of music, either Human voice, vocal or Musical instrument, instrumental, the musical form, structure of a musical piece or to the process of creating or writing a new pie ...

the four-group is the basic group of permutations in the twelve-tone technique
The twelve-tone technique—also known as dodecaphony, twelve-tone serialism, and (in British usage) twelve-note composition—is a method of musical composition
File:Chord chart.svg, 250px, Jazz and rock genre musicians may memorize the mel ...

. In that instance the Cayley table is written; Babbitt, Milton. (1960) "Twelve-Tone Invariants as Compositional Determinants", ''Musical Quarterly'' 46(2):253 Special Issue: Problems of Modern Music: The Princeton Seminar in Advanced Musical Studies (April): 246–59, Oxford University Press
Oxford University Press (OUP) is the university press
A university press is an academic publishing
Publishing is the activity of making information, literature, music, software and other content available to the public for sale or for fre ...

See also

*Quaternion group
In mathematics
Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). ...

* List of small groups
References

Further reading

* M. A. Armstrong (1988) ''Groups and Symmetry'',Springer Verlag
Springer Science+Business Media, commonly known as Springer, is a German multinational publishing
Publishing is the activity of making information, literature, music, software and other content available to the public for sale or for free. Tr ...

, page 53
Page most commonly refers to:
* Page (paper)
A page is one side of a leaf
A leaf (plural leaves) is the principal lateral appendage of the , usually borne above ground and specialized for . The leaves, stem, flower and fruit togethe ...

* W. E. Barnes (1963) ''Introduction to Abstract Algebra'', D.C. Heath & Co., page 20.
External links

* {{mathworld , urlname = Vierergruppe , title = Vierergruppe Finite groups