Klein four-group

In mathematics, the Klein four-group (or just Klein group or Viergruppe, often symbolized by the letter V), named after Felix Klein, is a group with four elements, the smallest non-cyclic group. Using the Sylow theorems, it can be easily proven that the only finite groups of order four, unique up to isomorphism, are the Klein four group, , and the cyclic group of order four, .

The Klein group's multiplication table is given by:

It may be visualized as the symmetry group of a rectangle:

   *************
   *           *
   *************

the four elements being: the identity, the vertical reflection, the horizontal reflection, and a 180 degree rotation.

All elements of the Klein group (except the identity) have order 2. It is abelian, and is isomorphic to C₂ × C₂, the direct product of two copies of the cyclic group of order 2. It is also isomorphic to the dihedral group of order 4.

The essential symmetry between the three elements of order 2 in the Klein four-group can be seen by its permutation representation on 4 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 A₄ (and also the symmetric group S₄) on 4 letters. According to Galois theory, the existence of the Klein four-group (and in particular, this particular representation) explains the existence of the formula for calculating the roots of quartic equations in terms of radicals.

One can also think of the Klein four-group as the automorphism group of the following graph:

  *   *
  |   
  *   *
      

The Klein four-group is the discrete part {1, j, −1, −j} of the group of units of the split-complex number ring.

Compare: quaternion group.