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Its multiplication table is given by:
| 1 | a | b | c | |
|---|---|---|---|---|
| 1 | 1 | a | b | c |
| a | a | 1 | c | b |
| b | b | c | 1 | a |
| c | c | b | a | 1 |
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:
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 equationsIn mathematics, a quartic equation is the result of setting a quartic function equal to zero. An example of a quartic equation is the equation : the general form is : As the fundamental theorem of algebra tells us, a quartic equation always has four solut in terms of radicals.One can also think of the Klein four-group as the automorphism group of the following graphGraph theory is the branch of mathematics that examines the properties of graphs . A graph with 6 vertices and 7 edges. Informally, a graph is a set of objects called vertices (or nodes) connected by links called edges (or arcs). Typically, a graph is dep:
* * | | * * | *The Klein four-group is the discrete part {1, j, -1, -j} of the group of units of the split-complex number ring.
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