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Vierergruppe - traduction vers Anglais
![4 element subsets of S<sub>4</sub>]]}}](https://commons.wikimedia.org/wiki/Special:FilePath/Klein four-group; Cayley table; subgroup of S4 (elements 0,1,6,7).svg?width=200 "4 element subsets of S<sub>4</sub>]]}}")
![transpositions]] of four objects form V](https://commons.wikimedia.org/wiki/Special:FilePath/Klein four-group; Cayley table; subgroup of S4 (elements 0,7,16,23).svg?width=200 "transpositions]] of four objects form V")
Wikipédia
In mathematics, the Klein four-group is a group with four elements, in which each element is self-inverse (composing it with itself produces the identity) and in which composing any two of the three non-identity elements produces the third one. It can be described as the symmetry group of a non-square rectangle (with the three non-identity elements being horizontal and vertical reflection and 180-degree rotation), as the group of bitwise exclusive or operations on two-bit binary values, or more abstractly as Z2 × Z2, the direct product of two copies of the cyclic group of order 2. It was named Vierergruppe (meaning four-group) by Felix Klein in 1884. It is also called the Klein group, and is often symbolized by the letter V or as K4.
The Klein four-group, with four elements, is the smallest group that is not a cyclic group. There is only one other group of order four, up to isomorphism, the cyclic group of order 4. Both are abelian groups.
