Klein group
In mathematics, the Klein four-group (or just Klein group or Vierergruppe, often symbolized by the letter V), named after Felix Klein, is a group with four elements, the smallest non-cyclic group.
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:
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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:
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Referenced By
Characteristic subgroup | Direct sum of groups | P-group | Permutation group
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