In mathematics, the Klein four-group is an abelian group with four elements, in which each element is 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 exclusive-or operations on two-bit binary values, or more abstractly as
Z
2
×
Z
2
{\displaystyle \mathbb {Z} _{2}\times \mathbb {Z} _{2}}
, the direct product of two copies of the cyclic group of order 2 by the Fundamental Theorem of Finitely Generated Abelian Groups. It was named Vierergruppe (German: [ˈfiːʁɐˌɡʁʊpə] , meaning four-group) by Felix Klein in 1884. It is also called the Klein group, and is often symbolized by the letter
V
{\displaystyle V}
or as
K
4
{\displaystyle K_{4}}
.
The Klein four-group, with four elements, is the smallest group that is not cyclic. Up to isomorphism, there is only one other group of order four: the cyclic group of order 4. Both groups are abelian.
Contents
Presentations
The Klein group's Cayley table is given by:
The Klein four-group is also defined by the group presentation
V
=
⟨
a
,
b
∣
a
2
=
b
2
=
(
a
b
)
2
=
e
⟩
.
{\displaystyle V=\left\langle a,b\mid a^{2}=b^{2}=(ab)^{2}=e\right\rangle .}
All non-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
D
4
{\displaystyle D_{4}}
Geometry
In two dimensions, the Klein four-group is the symmetry group of a rhombus and of rectangles that are not 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
D
2
{\displaystyle D_{2}}
one with a 2-fold rotation axis, and a perpendicular plane of reflection:
C
2
h
=
D
1
d
{\displaystyle C_{2\mathrm {h} }=D_{1\mathrm {d} }}
one with a 2-fold rotation axis in a plane of reflection (and hence also in a perpendicular plane of reflection):
C
2
v
=
D
1
h
{\displaystyle C_{2\mathrm {v} }=D_{1\mathrm {h} }}
Permutation representation
The three elements of order two in the Klein four-group are interchangeable: the automorphism group of V is thus the group of permutations of these three elements, that is, the symmetric group
S
3
{\displaystyle S_{3}}
.
The Klein four-group's permutations of its own elements can be thought of abstractly as its permutation representation on four points:
V
=
{\displaystyle V={}}
{(), (1,2)(3,4), (1,3)(2,4), (1,4)(2,3)}
In this representation,
V
{\displaystyle V}
is a normal subgroup of the alternating group
A
4
{\displaystyle A_{4}}
(and also the symmetric group
S
4
{\displaystyle S_{4}}
) on four letters. It is also a transitive subgroup of
S
4
{\displaystyle S_{4}}
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 equations in terms of radicals, as established by Lodovico Ferrari: the map
S
4
→
S
3
{\displaystyle S_{4}\to 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 four-group as their additive substructure.
If
R
×
{\displaystyle \mathbb {R} ^{\times }}
denotes the multiplicative group of non-zero reals and
R
+
{\displaystyle \mathbb {R} ^{+}}
the multiplicative group of positive reals, then
R
×
×
R
×
{\displaystyle \mathbb {R} ^{\times }\times \mathbb {R} ^{\times }}
Graph theory
Among the simple connected graphs, the one with the fewest entities admits the Klein four-group as its automorphism group is the diamond graph shown below. It is also the automorphism group of some other graphs with even 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, the four-group is the basic group of permutations in the twelve-tone technique. In that instance, the Cayley table is written
SATOR square
The SATOR square is a 5×5 word square :
It satisfies the following constraints :
A
:
(
i
,
j
)
=
(
j
,
i
)
{\displaystyle A:(i,j)=(j,i)}
B
:
(
i
,
j
)
=
(
6
−
i
,
6
−
j
)
{\displaystyle B:(i,j)=(6-i,6-j)}
Given those constraints,