Klein Four-Group

The Klein four-group is an abelian group with four elements, in which every element is its own inverse and 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, whose three non identity elements are horizontal reflection, vertical reflection and a 180 degree rotation, as the group of bitwise exclusive or operations on two bit binary values, or more abstractly as the direct product of two copies of the cyclic group of order two. It was named the Vierergruppe, German for four group, by Felix Klein in 1884, and is also called the Klein group, often written V or K4. With four elements it is the smallest group that is not cyclic; up to isomorphism it is one of only two groups of order four, the other being the cyclic group of order four, and both are abelian.