Klein four-group

Last updated December 04, 2024

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 $\mathbb {Z} _{2}\times \mathbb {Z} _{2}$ , 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.^[1] It is also called the Klein group, and is often symbolized by the letter $V$ or as $K_{4}$ .

Presentations

The Klein group's Cayley table is given by:

*	e	a	b	c
e	e	a	b	c
a	a	e	c	b
b	b	c	e	a
c	c	b	a	e

The Klein four-group is also defined by the group presentation

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 $D_{4}$ (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 direct sum $\mathbb {Z} _{2}\oplus \mathbb {Z} _{2}$ , so that it can be represented as the pairs {(0,0), (0,1), (1,0), (1,1)} under component-wise 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 four-group is thus an example of an 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 subsets of a powerset of a set with two elements—that is, over a field of sets with four elements, such as $\{\emptyset ,\{\alpha \},\{\beta \},\{\alpha ,\beta \}\}$ ; the empty set is the group's identity element in this case.

Another numerical construction of the Klein four-group 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).

The Klein four-group also has a representation as 2 × 2 real matrices with the operation being matrix multiplication:

e={\begin{pmatrix}1&0\\0&1\end{pmatrix}},\quad a={\begin{pmatrix}1&0\\0&-1\end{pmatrix}},\quad

b={\begin{pmatrix}-1&0\\0&1\end{pmatrix}},\quad c={\begin{pmatrix}-1&0\\0&-1\end{pmatrix}}

On a Rubik's Cube, the "4 dots" pattern can be made in three ways, 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

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. GreenRectangularCross.png — 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 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 $D_{2}$
one with a 2-fold rotation axis, and a perpendicular plane of reflection: $C_{2\mathrm {h} }=D_{1\mathrm {d} }$
one with a 2-fold rotation axis in a plane of reflection (and hence also in a perpendicular plane of reflection): $C_{2\mathrm {v} }=D_{1\mathrm {h} }$ .

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 $S_{3}$ .

The Klein four-group'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. It is also a transitive subgroup of $S_{4}$ that appears as a Galois group. In fact, it is the kernel of a surjective group homomorphism from $S_{4}$ to $S_{3}$ .

Other representations within S₄ are:

{ (), (1,2), (3,4), (1,2)(3,4) }

{ (), (1,3), (2,4), (1,3)(2,4) }

{ (), (1,4), (2,3), (1,4)(2,3) }

They are not normal subgroups of S₄.

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 $S_{4}\to S_{3}$ 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 $\mathbb {R} ^{\times }$ denotes the multiplicative group of non-zero reals and $\mathbb {R} ^{+}$ the multiplicative group of positive reals, then $\mathbb {R} ^{\times }\times \mathbb {R} ^{\times }$ is the group of units of the ring $\mathbb {R} \times \mathbb {R}$ , and $\mathbb {R} ^{+}\times \mathbb {R} ^{+}$ is a subgroup of $\mathbb {R} ^{\times }\times \mathbb {R} ^{\times }$ (in fact it is the component of the identity of $\mathbb {R} ^{\times }\times \mathbb {R} ^{\times }$ ). The quotient group $(\mathbb {R} ^{\times }\times \mathbb {R} ^{\times })/(\mathbb {R} ^{+}\times \mathbb {R} ^{+})$ 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

Among the simple connected graphs, the simplest (in the sense of having the fewest entities) that 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 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, the four-group is the basic group of permutations in the twelve-tone 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

Related Research Articles

In mathematics, an abelian group, also called a commutative group, is a group in which the result of applying the group operation to two group elements does not depend on the order in which they are written. That is, the group operation is commutative. With addition as an operation, the integers and the real numbers form abelian groups, and the concept of an abelian group may be viewed as a generalization of these examples. Abelian groups are named after Niels Henrik Abel.

A quotient group or factor group is a mathematical group obtained by aggregating similar elements of a larger group using an equivalence relation that preserves some of the group structure. For example, the cyclic group of addition modulo n can be obtained from the group of integers under addition by identifying elements that differ by a multiple of $and defining a group structure that operates on each such class as a single entity. It is part of the mathematical field known as group theory.$

In mathematics, given two groups, (G,∗) and (H, ·), a group homomorphism from (G,∗) to (H, ·) is a function h : G → H such that for all u and v in G it holds that

In abstract algebra, a group isomorphism is a function between two groups that sets up a bijection between the elements of the groups in a way that respects the given group operations. If there exists an isomorphism between two groups, then the groups are called isomorphic. From the standpoint of group theory, isomorphic groups have the same properties and need not be distinguished.

<span class="mw-page-title-main">Lie algebra</span> Algebraic structure used in analysis

In mathematics, a Lie algebra is a vector space $together with an operation called the Lie bracket, an alternating bilinear map, that satisfies the Jacobi identity. In other words, a Lie algebra is an algebra over a field for which the multiplication operation is alternating and satisfies the Jacobi identity. The Lie bracket of two vectors and is denoted . A Lie algebra is typically a non-associative algebra. However, every associative algebra gives rise to a Lie algebra, consisting of the same vector space with the commutator Lie bracket, .$

In mathematics, a group is a set with an operation that associates an element of the set to every pair of elements of the set and satisfies the following constraints: the operation is associative, it has an identity element, and every element of the set has an inverse element.

In abstract algebra, the symmetric group defined over any set is the group whose elements are all the bijections from the set to itself, and whose group operation is the composition of functions. In particular, the finite symmetric group $defined over a finite set of symbols consists of the permutations that can be performed on the symbols. Since there are such permutation operations, the order of the symmetric group is .$

In mathematics, specifically in group theory, the concept of a semidirect product is a generalization of a direct product. It is usually denoted with the symbol $⋉$ . There are two closely related concepts of semidirect product:

In abstract algebra, a cyclic group or monogenous group is a group, denoted C_n, that is generated by a single element. That is, it is a set of invertible elements with a single associative binary operation, and it contains an element g such that every other element of the group may be obtained by repeatedly applying the group operation to g or its inverse. Each element can be written as an integer power of g in multiplicative notation, or as an integer multiple of g in additive notation. This element g is called a generator of the group.

In mathematics, the endomorphisms of an abelian group X form a ring. This ring is called the endomorphism ring of X, denoted by End(X); the set of all homomorphisms of X into itself. Addition of endomorphisms arises naturally in a pointwise manner and multiplication via endomorphism composition. Using these operations, the set of endomorphisms of an abelian group forms a (unital) ring, with the zero map $as additive identity and the identity map as multiplicative identity.$

In mathematics, a dihedral group is the group of symmetries of a regular polygon, which includes rotations and reflections. Dihedral groups are among the simplest examples of finite groups, and they play an important role in group theory and geometry.

In group theory, the quaternion group Q₈ (sometimes just denoted by Q) is a non-abelian group of order eight, isomorphic to the eight-element subset $of the quaternions under multiplication. It is given by the group presentation$

In mathematics, the orthogonal group in dimension $n$ , denoted $O(n)$ , is the group of distance-preserving transformations of a Euclidean space of dimension $n$ that preserve a fixed point, where the group operation is given by composing transformations. The orthogonal group is sometimes called the general orthogonal group, by analogy with the general linear group. Equivalently, it is the group of $n \times n$ orthogonal matrices, where the group operation is given by matrix multiplication (an orthogonal matrix is a real matrix whose inverse equals its transpose). The orthogonal group is an algebraic group and a Lie group. It is compact.

In mathematics, a Cayley graph, also known as a Cayley color graph, Cayley diagram, group diagram, or color group, is a graph that encodes the abstract structure of a group. Its definition is suggested by Cayley's theorem, and uses a specified set of generators for the group. It is a central tool in combinatorial and geometric group theory. The structure and symmetry of Cayley graphs makes them particularly good candidates for constructing expander graphs.

In mathematics, a Fuchsian group is a discrete subgroup of PSL(2,R). The group PSL(2,R) can be regarded equivalently as a group of orientation-preserving isometries of the hyperbolic plane, or conformal transformations of the unit disc, or conformal transformations of the upper half plane, so a Fuchsian group can be regarded as a group acting on any of these spaces. There are some variations of the definition: sometimes the Fuchsian group is assumed to be finitely generated, sometimes it is allowed to be a subgroup of PGL(2,R), and sometimes it is allowed to be a Kleinian group which is conjugate to a subgroup of PSL(2,R).

In group theory, a field of mathematics, a double coset is a collection of group elements which are equivalent under the symmetries coming from two subgroups, generalizing the notion of a single coset.

In modular arithmetic, the integers coprime to n from the set $of n non-negative integers form a group under multiplication modulo n, called the multiplicative group of integers modulo n . Equivalently, the elements of this group can be thought of as the congruence classes, also known as residues modulo n, that are coprime to n . Hence another name is the group of primitive residue classes modulo n . In the theory of rings, a branch of abstract algebra, it is described as the group of units of the ring of integers modulo n . Here units refers to elements with a multiplicative inverse, which, in this ring, are exactly those coprime to n .$

In mathematics, D₃ (sometimes alternatively denoted by D₆) is the dihedral group of degree 3 and order 6. It equals the symmetric group S₃. It is also the smallest non-abelian group.

The Rubik's Cube group $represents the structure of the Rubik's Cube mechanical puzzle. Each element of the set corresponds to a cube move, which is the effect of any sequence of rotations of the cube's faces. With this representation, not only can any cube move be represented, but any position of the cube as well, by detailing the cube moves required to rotate the solved cube into that position. Indeed with the solved position as a starting point, there is a one-to-one correspondence between each of the legal positions of the Rubik's Cube and the elements of . The group operation is the composition of cube moves, corresponding to the result of performing one cube move after another.$

In mathematics, the special linear group SL(2, R) or SL₂(R) is the group of 2 × 2 real matrices with determinant one:

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) "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

External links

Weisstein, Eric W. "Vierergruppe". MathWorld .

This page is based on this Wikipedia article
Text is available under the CC BY-SA 4.0 license; additional terms may apply.
Images, videos and audio are available under their respective licenses.

[1] 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)

[2] 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

[1]

[2]