View source for Klein four-group

{{Short description|Mathematical abelian group}}
{{redirect|Vierergruppe|the four-person anti-Nazi Resistance group|Vierergruppe (German Resistance)}}
{{distinguish|text=[[Kleinian group]], a discrete subgroup of the Möbius transformations}}
{{Group theory sidebar |Finite}}

In [[mathematics]], the '''Klein four-group''' is an [[abelian group]] with four elements, in which each element is [[Involution (mathematics)|self-inverse]] (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]] of a non-square [[rectangle]] (with the three non-identity elements being [[horizontal reflection]], vertical reflection and [[180-degree rotation]]), as the group of [[bitwise operation|bitwise]] [[exclusive or|exclusive-or]] operations on two-bit binary values, or more [[abstract algebra|abstractly]] as <math>\mathbb{Z}_2\times\mathbb{Z}_2</math>, the [[Direct product of groups|direct product]] of two copies of the [[cyclic group]] of [[Order (group theory)|order]] 2 by the [[Fundamental theorem of finitely generated abelian groups|Fundamental Theorem of Finitely Generated Abelian Groups]]. It was named '''''Vierergruppe''''' ({{IPA|de|ˈfiːʁɐˌɡʁʊpə|lang|De-Vierergruppe.ogg}}, meaning four-group) by [[Felix Klein]] in 1884.<ref>''Vorlesungen über das Ikosaeder und die Auflösung der Gleichungen vom fünften Grade'' (Lectures on the icosahedron and the solution of equations of the fifth degree)</ref> It is also called the '''Klein group''', and is often symbolized by the letter <math>V</math> or as <math>K_4</math>.

The Klein four-group, with four elements, is the smallest [[Group (mathematics)|group]] that is not cyclic. Up to [[Group isomorphism|isomorphism]], there is only one other group of order four: the cyclic group of order 4. Both groups are abelian.

== Presentations ==
The Klein group's [[Cayley table]] is given by:
{| class=wikitable width=120
!*
!e
!''a''
!''b''
!''c''
|- align=center
!e
|e || ''a'' || ''b'' || ''c''
|- align=center
!''a''
| ''a'' || e || ''c'' || ''b''
|- align=center
!''b''
|''b'' || ''c'' || e || ''a''
|- align=center
!''c''
| ''c'' || ''b'' || ''a'' || e
|}

The Klein four-group is also defined by the [[presentation of a group|group presentation]]
: <math>V = \left\langle a,b \mid a^2 = b^2 = (ab)^2 = e \right\rangle.</math>

All non-[[identity element|identity]] elements of the Klein group have order 2, so any two non-identity elements can serve as generators in the above presentation. The Klein four-group is the smallest non-[[cyclic group]]. It is, however, an [[abelian group]], and isomorphic to the [[dihedral group]] of order (cardinality) 4, symbolized <math>D_4</math> (or <math>D_2</math>, 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]] <math>\mathbb{Z}_2\oplus\mathbb{Z}_2</math>, so that it can be represented as the pairs {{nowrap|{(0,0), (0,1), (1,0), (1,1)} }} under component-wise addition [[Modular arithmetic|modulo 2]] (or equivalently the [[Bit array|bit strings]] {{nowrap|{00, 01, 10, 11} }}under [[bitwise XOR]]), with (0,0) being the group's identity element. The Klein four-group is thus an example of an [[elementary abelian group|elementary abelian 2-group]], which is also called a [[Boolean group]]. The Klein four-group is thus also the group generated by the [[symmetric difference]] as the binary operation on the [[subset]]s of a [[powerset]] of a set with two elements—that is, over a [[field of sets]] with four elements, such as <math>\{\emptyset,\{\alpha\},\{\beta\},\{\alpha,\beta\}\}</math>; the [[empty set]] is the group's identity element in this case.

Another numerical construction of the Klein four-group is the set {{nowrap|{{mset| 1, 3, 5, 7 }}}}, with the operation being [[Multiplicative group of integers modulo n|multiplication modulo 8]]. Here ''a'' is 3, ''b'' is 5, and {{nowrap|1=''c'' = ''ab''}} is {{nowrap|1=3 × 5 = 15 ≡ 7 (mod 8)}}.

The Klein four-group also has a representation as {{nowrap|2 × 2}} real matrices with the operation being matrix multiplication:
: <math>
  e =\begin{pmatrix}
      1 & 0\\
      0 & 1
    \end{pmatrix},\quad
  a = \begin{pmatrix}
      1 &  0\\
      0 & -1
    \end{pmatrix},\quad
</math>
: <math>
  b = \begin{pmatrix}
      -1 & 0\\
       0 & 1
    \end{pmatrix},\quad
  c = \begin{pmatrix}
      -1 &  0\\
       0 & -1
    \end{pmatrix}
</math>

On a [[Rubik's Cube]], the "4 dots" pattern can be made in three ways (for example, M2 U2 M2 U2 F2 M2 F2), depending on the pair of faces that are left blank; these three positions together with the solved position form an example of the Klein group, with the solved position serving as the identity.

== Geometry ==
[[File:GreenRectangularCross.png|thumb|right|''V'' is the symmetry group of this cross: flipping it horizontally (''a'') or vertically (''b'') or both (''ab'') leaves it unchanged.  A quarter-turn changes it.]]
In two dimensions, the Klein four-group is the [[symmetry group]] of a [[rhombus]] and of [[rectangle]]s that are not [[square (geometry)|squares]], the four elements being the identity, the vertical reflection, the horizontal reflection, and a 180° rotation.

In three dimensions, there are three different symmetry groups that are algebraically the Klein four-group:
* one with three perpendicular 2-fold rotation axes: the [[dihedral group]] <math>D_2</math>
* one with a 2-fold rotation axis, and a perpendicular plane of reflection: <math>C_{2\mathrm{h}}=D_{1\mathrm{d}}</math>
* one with a 2-fold rotation axis in a plane of reflection (and hence also in a perpendicular plane of reflection): <math>C_{2\mathrm{v}}=D_{1\mathrm{h}}</math>.

== Permutation representation ==
[[File:Klein four-group; Cayley table; subgroup of S4 (elements 0,7,16,23).svg|thumb|The identity and double-[[transposition (mathematics)|transpositions]] of four objects form ''V''.]]
[[File:Klein four-group; Cayley table; subgroup of S4 (elements 0,1,6,7).svg|thumb|Other permutations of four objects can form ''V'' as well.{{paragraph}}{{hatnote|See [[Subgroup#4 elements|4 element subsets of S<sub>4</sub>]]}}]]
The three elements of order two in the Klein four-group are interchangeable: the [[automorphism group]] of ''V'' is thus the group of [[Permutation|permutations]] of these three elements, that is, the [[symmetric group]] <math>S_3</math>.

The Klein four-group's permutations of its own elements can be thought of abstractly as its [[permutation representation]] on four points:
:<math>V={}</math>{(), (1,2)(3,4), (1,3)(2,4), (1,4)(2,3)}

In this representation, <math>V</math> is a [[normal subgroup]] of the [[alternating group]] <math>A_4</math> (and also the [[symmetric group]] <math>S_4</math>) on four letters. It is also a transitive subgroup of <math>S_4</math>that appears as a [[Galois group]]. In fact, it is the [[Kernel (algebra)#Group homomorphisms|kernel]] of a surjective [[group homomorphism]] from <math>S_4</math> to <math>S_3</math>.

Other representations within ''S''<sub>4</sub> are:
: {{mset| (), (1,2), (3,4), (1,2)(3,4) }}
: {{mset| (), (1,3), (2,4), (1,3)(2,4) }}
: {{mset| (), (1,4), (2,3), (1,4)(2,3) }}

They are not normal subgroups of ''S''<sub>4</sub>.
{{clear}}

== Algebra ==
According to [[Galois theory]], 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]]s in terms of [[Radical of an algebraic group|radical]]s, as established by [[Lodovico Ferrari]]: the map <math>S_4\to S_3</math> corresponds to the [[resolvent cubic]], in terms of [[Lagrange resolvents]].

In the construction of [[finite ring]]s, eight of the eleven rings with four elements have the Klein four-group as their additive substructure.

If <math>\mathbb{R}^{\times}</math> denotes the multiplicative group of non-zero reals and <math>\mathbb{R}^+</math> the multiplicative group of [[positive reals]], then <math>\mathbb{R}^{\times}\times\mathbb{R}^{\times}</math> is the [[group of units]] of the ring <math>\mathbb{R}\times\mathbb{R}</math>, and <math>\mathbb{R}^+\times\mathbb{R}^+</math> is a subgroup of <math>\mathbb{R}^{\times}\times\mathbb{R}^{\times}</math> (in fact it is the [[identity component|component of the identity]] of <math>\mathbb{R}^{\times}\times\mathbb{R}^{\times}</math>). The [[quotient group]] <math>(\mathbb{R}^{\times}\times\mathbb{R}^{\times})/(\mathbb{R}^+\times\mathbb{R}^+)</math> is isomorphic to the Klein four-group. In a similar fashion, the group of units of the [[split-complex number|split-complex number ring]], when divided by its identity component, also results in the Klein four-group.

== Graph theory ==
Among the [[simple graph|simple]] [[connected graph]]s, the simplest (in the sense of having the fewest entities) that admits the Klein four-group as its [[graph automorphism|automorphism group]] is the [[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.

{{multiple image|align = center|perrow = 3
| image1 = Diamond graph.svg  | width1=150 | height1=150
| image2 = Klein 4-Group Graph.svg  | width2=150 | height2=150
| image3 = Digon graph.svg | width3=338 | height3=150
}}

== Music ==
In [[music composition]], the four-group is the basic group of permutations in the [[twelve-tone technique]]. In that instance, the Cayley table is written<ref>[[Milton Babbitt|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]]</ref>
{| class=wikitable
! S !! I !! R !! RI
|-
! I
| S || RI || R
|-
! R
| RI || S || I
|-
! RI
| R || I || S
|}

== See also ==
* [[Quaternion group]]
* [[List of small groups]]

== References ==
{{reflist}}

== Further reading ==
* M. A. Armstrong (1988) ''Groups and Symmetry'', [[Springer Verlag]], [{{Google books|plainurl=y|id=lDfcBPl9Cu4C|page=53|text=Klein's group}} page 53].
* W. E. Barnes (1963) ''Introduction to Abstract Algebra'', D.C. Heath & Co., page 20.

== External links ==
* {{mathworld | urlname = Vierergruppe  | title = Vierergruppe }}

[[Category:Finite groups]]