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 ${\displaystyle \mathbb {Z} _{2}\times \mathbb {Z} _{2}}$, 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 ${\displaystyle V}$ or as ${\displaystyle K_{4}}$.

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.

The Klein group's Cayley table is given by:

The Klein four-group is also defined by the group presentation

All non-identity elements of the Klein group have order 2, so any two non-identity elements can serve as generators in the above presentation. The Klein four-group is the smallest non-cyclic group. It is, however, an abelian group, and isomorphic to the dihedral group of order (cardinality) 4, symbolized ${\displaystyle D_{4}}$ (or ${\displaystyle D_{2}}$, using the geometric convention); other than the group of order 2, it is the only dihedral group that is abelian.

The Klein four-group is also isomorphic to the direct sum ${\displaystyle \mathbb {Z} _{2}\oplus \mathbb {Z} _{2}}$, so that it can be represented as the pairs {(0,0), (0,1), (1,0), (1,1)} under component-wise addition modulo 2 (or equivalently the bit strings {00, 01, 10, 11} under bitwise XOR), with (0,0) being the group's identity element. The Klein four-group is thus an example of an elementary abelian 2-group, which is also called a Boolean group. The Klein four-group is thus also the group generated by the symmetric difference as the binary operation on the subsets of a powerset of a set with two elements—that is, over a field of sets with four elements, such as ${\displaystyle \{\emptyset ,\{\alpha \},\{\beta \},\{\alpha ,\beta \}\}}$; the empty set is the group's identity element in this case.

Another numerical construction of the Klein four-group is the set { 1, 3, 5, 7 }, with the operation being multiplication modulo 8. Here a is 3, b is 5, and c = ab is 3 × 5 = 15 ≡ 7 (mod 8).

The Klein four-group also has a representation as 2 × 2 real matrices with the operation being matrix multiplication: