Klein four-group

Group theory

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In mathematics, the Klein four-group (or just Klein group or Vierergruppe, often symbolized by the letter V) 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 his Vorlesungen über das Ikosaeder und die Auflösung der Gleichungen vom fünften Grade in 1884.

The Klein four-group is the smallest non-cyclic group. The only other group with four elements, up to isomorphism, is Z₄, the cyclic group of order four (see also the list of small groups).

All non-identity elements of the Klein group have order 2. It is abelian, and isomorphic to the dihedral group of order (cardinality) 4. It is also isomorphic to the direct sum :

The Klein group's Cayley table is given by:

*	1	a	b	ab
1	1	a	b	ab
a	a	1	ab	b
b	b	ab	1	a
ab	ab	b	a	1

In 2D it is the symmetry group of a rhombus and of a rectangle, 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 which 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 is the group of permutations of the three elements. This essential symmetry can also be seen by its permutation representation on 4 points:

V = { identity, (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 map 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 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:

The Klein four-group is the group of components of the group of units of the topological ring of split-complex numbers.

Another example of the Klein four-group is the multiplicative group { 1, 3, 5, 7 } with the action being multiplication modulo 8.

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

See also

• Quaternion group
• Kleinian group

References

• M.A. Armstrong (1988) Groups and Symmetry, Springer Verlag, page 53.
• W.E. Barnes (1963) Introduction to Abstract Algebra, D.C. Heath & Co., page 20.
• Weisstein, Eric W., "Klein Four-group" from MathWorld.