In mathematics, the Klein four-group is an abelian group with four elements, in which each element is self-inverse 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, 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 2 by the Fundamental Theorem of Finitely Generated Abelian Groups. It was named Vierergruppe by Felix Klein in 1884. It is also called the Klein group, and is often symbolized by the letter or as .