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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 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 ...

rectangle
(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 ...

cyclic group
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 K4. The Klein four-group, with four elements, is the smallest group that is not a
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 ...

cyclic group
. 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's
Cayley 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 ...
:\mathrm = \left\langle a,b \mid a^2 = b^2 = (ab)^2 = e \right\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 ...

cyclic group
. 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. D4 (or D2, 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
direct 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 ...

subset
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 ...

powerset
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. \; 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 ...

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, ''b'' is 5, and is . The Klein four-group has a representation as 2×2 real matrices with the operation being matrix multiplication: : e =\begin 1 & 0\\ 0 & 1 \end,\quad a = \begin 1 & 0\\ 0 & -1 \end,\quad b = \begin -1 & 0\\ 0 & 1 \end,\quad c = \begin -1 & 0\\ 0 & -1 \end


Geometry

Geometrically, in two dimensions the Klein four-group is 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
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 ...

rhombus
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 ...

rectangle
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 ...

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 (and hence also in a perpendicular plane of reflection): .


Permutation representation

The three elements of order two in the Klein four-group are interchangeable: the
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 ...
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 ...
A4 (and also the
symmetric 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 ...
S4) on four letters. In fact, it is the
kernel 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 ...

group homomorphism
from S4 to S3. Other representations within S4 are: : : : They are not normal subgroups of S4.


Algebra

According to
Galois 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 R× denotes the multiplicative group of non-zero reals and R+ the multiplicative group of positive reals, R× × R× is the
group 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 simplest
simple 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 ...

simple
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 ...

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.


Music

In
music 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 ...

Oxford University Press


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