Klein four-group

This article is about the mathematical concept. For the four-person anti-Nazi Resistance groups, see Vierergruppe (German Resistance).

Not to be confused with Kleinian group, a discrete subgroup of PSL(2, C)..

The symmetry group of this cross is the Klein four-group. It can be flipped horizontally (a) or vertically (b) or both (ab) and remain unchanged. Unlike a square, though, a quarter-turn rotation will change the figure.

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. 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:

File:Klein four-group; Cayley table; subgroup of S4 (elements 0,7,16,23).svg

Identity and double-transpositions of four objects form V

File:Klein four-group; Cayley table; subgroup of S4 (elements 0,1,6,7).svg

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:

50px

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;

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

See also

• Quaternion group

Further reading

• 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.

External Links

• Weisstein, Eric W., "Vierergruppe", MathWorld.

• Vorlesungen über das Ikosaeder und die Auflösung der Gleichungen vom fünften Grade (Lectures on the icosahedron and the solution of equations of the fifth degree)
• Babbitt, Milton. (1960) "Twelve-Tone Invariants as Compositional Determinants", Musical Quarterly 46(2):253 Special Issue: Problems of Modern Music: The Princeton Seminar in Advanced Musical Studies (April): 246–59, Oxford University Press