Klein four-group

Mathematical abelian group In mathematics, the Klein four-group is an abelian 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 reflection, vertical reflection and 180-degree rotation), as the group of bitwise exclusive-or operations on two-bit binary values, or more abstractly as Z 2 × Z 2 {\displaystyle \mathbb {Z} _{2}\times \mathbb {Z} _{2}} , the direct product of two copies of the cyclic group of order 2 by the Fundamental Theorem of Finitely Generated Abelian Groups. It was named Vierergruppe (German: [ˈfiːʁɐˌɡʁʊpə] ⓘ, meaning four-group) by Felix Klein in 1884. It is also called the Klein group, and is often symbolized by the letter V {\displaystyle V} or as K 4 {\displaystyle K_{4}} . The Klein four-group, with four elements, is the smallest group that is not cyclic. Up to isomorphism, there is only one other group of order four: the cyclic group of order 4. Both groups are abelian. Present