Klein four-group

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.^[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 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, 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 two dimensions 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 three dimensions 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:

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

See also

• Quaternion group

References

1. 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)
2. 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

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.