Klein four-group

In group theory is the Klein four-group, also briefly called Group of Four, the smallest non- cyclic group. She has the group order 4, as only the cyclic group beside her, and as this is an abelian group. It was named after Felix Klein, in his Lectures on the icosahedron and the solution of equations of the fifth degree of this group said, as the " Group of Four " in 1884; as a symbol often used the letter. The group of four is not characterized by a special presentation of its elements, but of an abstract and corresponds to the finite group.

Truth table

The Klein's four group operates on an underlying set of cardinality ( cardinality) 4 and has four elements, for example, one of which is the neutral element. Their (internal ) link together again yields one of the elements in the wrong order each linked pairs the same result ( commutative ), with (two-digit ) linking an element with itself ever the neutral element, and is given by the following truth table:

This panel of two -digit shortcut is symmetric with respect, as in all commutative groups the main diagonal, which in the group of four - is occupied solely by the neutral element - unlike, for example, in the cyclic group of the same order (C4). Thus, each element is also to be (two-sided ) inverse element or involutive.

The copies of the header and input line to find at conventional notation as here in the 1st row and the 1st column, identify the ( on both sides ) neutral element, which is called as identity mapping of the elements and "identity".

Properties

The Klein four-group is a commutative, but no cyclic group. Their subgroups are { 1}, {1, a}, {1, b}, {1, ab}, {1, a, b, ab} and all normal, the foursome is thus no finite simple group. The non-neutral elements, the elements have order 2, each element forms its own conjugacy class.

The group of four corresponds to the ( abelian and non- cyclic ) finite group - a direct product of two copies of the cyclic group, which is the smallest non -trivial group and the group only order 2. The abstract properties of the group of four can be shown on the example of different point groups and multiplicative groups which are isomorphic to her.

Representations

The foursome occur, for example as the symmetry group of a non- equiangular rhombus or a non- equilateral rectangle ( which are therefore not a square, whose symmetry group would be the dihedral group ( the group order 8 ) and the rotation group of a square is an example of the cyclic group ):

The four elements are: the identity (or 0 ° rotation ), as the reflection of the central perpendicular axis than the reflection at the horizontal center axis, and when the 180 ° rotation around the center (which horizontal as combined and can be considered vertical mirroring). The labeling of the corners of the rectangle above provides the permutation

And with notation of permutations in Zykelschreibweise

In this representation, the commutator subgroup and thus a normal subgroup of the alternating group and a normal subgroup of the symmetric group. In the Galois theory the existence of the Klein four-group explained in that the existence of the solution formula for equations of the fourth degree.

Furthermore, the group of four isomorphic to

  • ,
  • The dihedral group of order 4 ()
  • The unit group of the ring (that is the residue classes of 1, 3, 5 and 7 multiplication modulo 8),
  • The unit group of the ring (that is the residue classes of 1, 5, 7 and 11 with multiplication modulo 12),
  • The automorphism group of the following graphs:
  • The and of the involutions with any body

Generated group with the sequential execution as a group link.

Documents

Groups of small order

  • The finite set
  • Group Theory
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