Klein fourgroup
Algebraic structure → Group theory Group theory 


Modular groups

Infinite dimensional Lie group

In mathematics, the Klein fourgroup (or just Klein group or Vierergruppe (English: fourgroup), often symbolized by the letter V or as K_{4}) is the group Z_{2} × Z_{2}, the direct product of two copies of the cyclic group of order 2. It was named Vierergruppe by Felix Klein in 1884.^{[1]}
With four elements, the Klein fourgroup is the smallest noncyclic group, and every noncyclic group of order 4 is isomorphic to the Klein fourgroup. The cyclic group of order 4 and the Klein fourgroup are therefore, up to isomorphism, the only groups of order 4. Both are abelian groups. The smallest nonabelian group is the symmetric group of degree 3, which has order 6.
Presentations
The Klein group's Cayley table is given by:
*  1  a  b  c 

1  1  a  b  c 
a  a  1  c  b 
b  b  c  1  a 
c  c  b  a  1 
The Klein fourgroup is also defined by the group presentation
All nonidentity elements of the Klein group have order 2, thus any two nonidentity elements can serve as generators in the above presentation. The Klein fourgroup is the smallest noncyclic group. It is however an abelian group, and isomorphic to the dihedral group of order (cardinality) 4, Dih_{2}; other than the group of order 2, it is the only dihedral group that is abelian.
The Klein fourgroup is also isomorphic to the direct sum Z_{2} ⊕ Z_{2}, so that it can be represented as the pairs {(0,0), (0,1), (1,0), (1,1)} under componentwise addition modulo 2 (or equivalently the bit strings {00, 01, 10, 11} under bitwise XOR); with (0,0) being the group's identity element. The Klein fourgroup is thus an example of an elementary abelian 2group, which is also called a Boolean group. The Klein fourgroup is thus also the group generated by the symmetric difference as the binary operation on the subsets of a powerset of a set with two elements, i.e. over a field of sets with four elements, e.g. ; the empty set is the group's identity element in this case.
Another numerical construction of the Klein fourgroup is the set { 1, 3, 5, 7 }, with the operation being multiplication modulo 8. Here a is 3, b is 5, and c = ab is 3 × 5 = 15 ≡ 7 (mod 8).
Geometry
Geometrically, in two dimensions the Klein fourgroup is the symmetry group of a rhombus and of a rectangle which are not 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 fourgroup V:
 one with three perpendicular 2fold rotation axes: D_{2}
 one with a 2fold rotation axis, and a perpendicular plane of reflection: C_{2h} = D_{1d}
 one with a 2fold rotation axis in a plane of reflection (and hence also in a perpendicular plane of reflection): C_{2v} = D_{1h}.
Permutation representation
The three elements of order two in the Klein fourgroup are interchangeable: the automorphism group of V is the group of permutations of these three elements.
The Klein fourgroup's permutations of its own elements can be thought of abstractly as its permutation representation on four points:
 V = { (), (1,2)(3,4), (1,3)(2,4), (1,4)(2,3) }
In this representation, V is a normal subgroup of the alternating group A_{4} (and also the symmetric group S_{4}) on four letters. In fact, it is the kernel of a surjective group homomorphism from S_{4} to S_{3}.
Algebra
According to Galois theory, the existence of the Klein fourgroup (and in particular, the permutation representation of it) explains the existence of the formula for calculating the roots of quartic equations in terms of radicals, as established by Lodovico Ferrari: the map S_{4} → S_{3} corresponds to the resolvent cubic, in terms of Lagrange resolvents.
In the construction of finite rings, eight of the eleven rings with four elements have the Klein fourgroup as their additive substructure.
If R^{×} denotes the multiplicative group of nonzero reals and R^{+} the multiplicative group of positive reals, R^{×} × R^{×} is the group of units of the ring R × R, and R^{+} × R^{+} is a subgroup of R^{×} × R^{×} (in fact it is the component of the identity of R^{×} × R^{×}). The quotient group (R^{×} × R^{×}) / (R^{+} × R^{+}) is isomorphic to the Klein fourgroup. In a similar fashion, the group of units of the splitcomplex number ring, when divided by its identity component, also results in the Klein fourgroup.
Graph theory
The Klein fourgroup as a subgroup of the alternating group A_{4} is not the automorphism group of any simple graph. It is, however, the automorphism group of a twovertex graph where the vertices are connected to each other with two edges, making the graph nonsimple. 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 topleft, bottomleft, topright, bottomright:
Music
In music composition the fourgroup is the basic group of permutations in the twelvetone technique. In that instance the Cayley table is written;^{[2]}
S  I:  R:  RI: 
I:  S  RI  R 
R:  RI  S  I 
RI:  R  I  S 
In popular culture
The Klein Four Group is the name of an a cappella group made up of five mathematics graduate students at Northwestern University, best known for their song Finite Simple Group (of Order Two)
See also
References
 ↑ 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)
 ↑ Babbitt, Milton. (1960) "TwelveTone 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
Further reading
 M. A. Armstrong (1988) Groups and Symmetry, Springer Verlag, page 53.
 W. E. Barnes (1963) Introduction to Abstract Algebra, D.C. Heath & Co., page 20.