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In mathematics, the Klein four-group (or just Klein group or Vierergruppe (English: four-group), often symbolized by the letter V or as K₄) is the group Z₂ × Z₂, the direct product of two copies of the cyclic group of order 2. It was named Vierergruppe by Felix Klein in 1884.^[1] The Klein group's Cayley table is given by::

*	1	a	b	c
1	1	a	b	c
a	a	1	c	b
b	b	c	1	a
c	c	b	a	1

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

\( \mathrm{V} = \langle a,b \mid a^2 = b^2 = (ab)^2 = 1 \rangle. \)

All non-identity elements of the Klein group have order 2, thus 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; it is the only dihedral group that is abelian.

The Klein four-group is also isomorphic to the direct sum Z2 ⊕ Z2, so that it can be represented as the bit vectors {00, 01, 10, 11} under bitwise XOR (or equivalently under bitwise addition modulo 2); note that 00 is the group's identity element in this case. 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, i.e. over a field of sets with four elements, e.g. \( \{\emptyset, \{\alpha\}, \{\beta\}, \{\alpha, \beta\}\} \) ; 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).

Geometrically, in 2D the Klein four-group is the symmetry group of a rhombus and of a rectangle which are not squares, the four elements being the identity, the vertical reflection, the horizontal reflection, and a 180 degree rotation.

In 3D there are three different symmetry groups that are algebraically the Klein four-group V:

one with three perpendicular 2-fold rotation axes: D₂
one with a 2-fold rotation axis, and a perpendicular plane of reflection: C_2h = D_1d
one with a 2-fold rotation axis in a plane of reflection (and hence also in a perpendicular plane of reflection): C_2v = D_1h.

The three elements of order 2 in the Klein four-group are interchangeable: the automorphism group of V is the group of permutations of these three elements.

The Klein four-group's permutations of its own elements can be thought of abstractly as its permutation representation on 4 points:
Identity and double-transpositions of four objects form V
Other permutations of four objects, forming V as well

See: 4 element subsets of S4

V = { (), (1,2)(3,4), (1,3)(2,4), (1,4)(2,3) }

In this representation, V is a normal subgroup of the alternating group A₄ (and also the symmetric group S₄) on 4 letters. In fact, it is the kernel of a surjective group homomorphism from S₄ to S₃. According to Galois theory, the existence of the Klein four-group (and in particular, this representation of it) explains the existence of the formula for calculating the roots of quartic equations in terms of radicals, as established by Lodovico Ferrari: the map S₄ → S₃ corresponds to the resolvent cubic, in terms of Lagrange resolvents.

The Klein four-group as a subgroup of A₄ is not the automorphism group of any simple graph. It is, however, the automorphism group of a two-vertex graph where the vertices are connected to each other with two edges, making the graph non-simple. It is also the automorphism group of the following simple graph, but in the permutation representation { (), (1,2), (3,4), (1,2)(3,4) } where the points are labeled top-left, bottom-left, top-right, bottom-right:

Klein 4-Group Graph

In the construction of finite rings, eight of the eleven rings with four elements have the Klein four-group as their additive substructure.

If R× denotes the multiplicative group of non-zero reals and R+ the multiplicative group of positive reals, R× × R× is the group of units of the ring R×R and R+ × R+ is a subgroup of R× × R× (in fact it is the component of the identity of R× × R×). The quotient group (R× × R×) / (R+ × R+) is isomorphic to the Klein four-group. In a similar fashion, the group of units of the split-complex number ring, when divided by its identity component, also results in the Klein four-group.

In music composition the four-group is the basic group of permutations in the twelve-tone technique. In that instance the Cayley table is written;[2]

S	I:	R:	RI:
I:	S	RI	R
R:	RI	S	I
RI:	R	I	S