In mathematics, the Klein four-group (or just Klein group or Vierergruppe, often symbolized by the letter V) is the group Z2 × Z2, the direct product of two copies of the cyclic group of order 2 (or any isomorphic variant). Mathematics is the body of Knowledge and Academic discipline that studies such concepts as Quantity, Structure, Space and In Mathematics, a group is a set of elements together with an operation that combines any two of its elements to form a third element In Mathematics, one can often define a direct product of objectsalready known giving a new one In Group theory, a cyclic group or monogenous group is a group that can be generated by a single element in the sense that the group has an In Group theory, a branch of Mathematics, the term order is used in two closely related senses the order of a group is In Abstract algebra, an isomorphism ( Greek: ἴσος isos "equal" and μορφή morphe "shape" is a bijective It was named Vierergruppe by Felix Klein in his Vorlesungen über das Ikosaeder und die Auflösung der Gleichungen vom fünften Grade in 1884. Felix Christian Klein ( 25 April 1849 &ndash 22 June 1925) was a German Mathematician, known for his work in Group
The Klein four-group is the smallest non-cyclic group. The only other group with four elements, up to isomorphism, is the cyclic group of order four: Z4 (see also the list of small groups). The following list in Mathematics contains the Finite groups of small order Up to Group isomorphism.
All elements of the Klein group (except the identity) have order 2. In Mathematics, an identity element (or neutral element) is a special type of element of a set with respect to a Binary operation on that It is abelian, and isomorphic to the dihedral group of order 4. An abelian group, also called a commutative group, is a group satisfying the additional requirement that the product of elements does not depend on their order (the In Mathematics, a dihedral group is the group of symmetries of a Regular polygon, including both rotations and reflections
The Klein group's Cayley table is given by:
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* 1 i j k 1 1 i j k i i 1 k j j j k 1 i k k j i 1
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In 2D it is the symmetry group of a rhombus and of a rectangle, the four elements being the identity, the vertical reflection, the horizontal reflection, and a 180 degree rotation. A Cayley table, after the 19th century British Mathematician Arthur Cayley, describes the structure of a The Symmetry group of an object ( Image, signal, etc eg in 1D 2D or 3D is the group of all Isometries under which it is In Geometry, a rhombus (from Ancient Greek ῥόμβος - rrhombos “rhombus spinning top” (plural rhombi or rhombuses In Geometry, a rectangle is defined as a Quadrilateral where all four of its angles are Right angles A rectangle with vertices ABCD would be denoted as
In 3D there are three different symmetry groups which 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: C2h = D1d
- one with a 2-fold rotation axis in a plane of reflection (and hence also in a perpendicular plane of reflection): C2v = D1h
The three elements of order 2 in the Klein four-group are interchangeable: the automorphism group is the group of permutations of the three elements. In Mathematics, an automorphism is an Isomorphism from a Mathematical object to itself This essential symmetry can also be seen by its permutation representation on 4 points:
- V = { identity, (1,2)(3,4), (1,3)(2,4), (1,4)(2,3) }
In this representation, V is a normal subgroup of the alternating group A4 (and also the symmetric group S4) on 4 letters. In Algebra and Geometry, a group action is a way of describing symmetries of objects using groups. In Mathematics, more specifically in Abstract algebra, a normal subgroup is a special kind of Subgroup. In Mathematics, an alternating group is the group of Even permutations of a Finite set. In Mathematics, the symmetric group on a set X, denoted by S X or Sym( X) is the group whose underlying In fact, it is the kernel of a surjective map from S4 to S3. According to Galois theory, the existence of the Klein four-group (and in particular, this representation of it) explains the existence of the formula for calculating the roots of quartic equations in terms of radicals. In Mathematics, more specifically in Abstract algebra, Galois theory, named after Évariste Galois, provides a connection between field theory In Mathematics, a quartic equation is one which can be expressed as a Quartic function equalling zero
The Klein four-group as a subgroup of A4 is not the automorphism group of any simple graph. In Mathematics, an automorphism is an Isomorphism from a Mathematical object to itself In Mathematics and Computer science, graph theory is the study of graphs: mathematical structures used to model pairwise relations between objects It is, however, the automorphism group of a two-vertex graph where the vertices are connected to each other with two edges, making the graph non-simple. 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 top-left, bottom-left, top-right, bottom-right:
The Klein four-group is the group of components of the group of units of the topological ring of split-complex numbers. In Mathematics, the identity component of a Topological group G is the connected component G 0 that contains the Identity In Mathematics, a unit in a ( Unital) ring R is an invertible element of R, i In Mathematics, a topological ring is a ring R which is also a Topological space such that both the addition and the multiplication are In Linear algebra, a split-complex number is of the form z = x + y j where j2 = +1, and x and y are Real
Another example of the Klein four-group is the multiplicative group { 1, 3, 5, 7 } with the action being multiplication modulo 8. In Mathematics and Group theory the term multiplicative group refers to one of the following concepts depending on the context any group \scriptstyle\mathfrak
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Field
The Klein four-group is isomorphic to the additive group of finite field GF(4):
+ | 0 1 A B · | 0 1 A B --+-------- --+-------- 0 | 0 1 A B 0 | 0 0 0 0 1 | 1 0 B A 1 | 0 1 A B A | A B 0 1 A | 0 A B 1 B | B A 1 0 B | 0 B 1 A
In Popular Culture
The Klein Four is an a cappella singing group at Northwestern University, best known for their song "Finite SImple Group (of Order Two). In Abstract algebra, a finite field or Galois field (so named in honor of Évariste Galois) is a field that contains only finitely many elements A cappella (Italian or Latin "From the chapel/choir" Music is Vocal music or Singing without instrumental Accompaniment "[1]
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In Mathematics, a dihedral group is the group of symmetries of a Regular polygon, including both rotations and reflections In Group theory, the quaternion group is a non-abelian group of order 8 In Mathematics, a Kleinian group, named after Felix Klein, is a finitely generated Discrete group &Gamma of orientation preserving conformalDapyx Software network: MP3 Explorer | Ebook Manager | Zenithic