Klein group of four

This article does not deal with the Klein groups , special groups of Möbius transformations, but the non-cyclic group with 4 elements.

In group theory , Klein's group of four , also called group of four for short , is the smallest non-cyclic group . It has group order 4, like only the cyclic group next to it, and like this it is an Abelian group . It bears its name after Felix Klein , who in 1884 spoke of this group as a "group of four" in his lectures on the icosahedron and the solution of the equations of the fifth degree . The letter is often used as a symbol . The group of four is not characterized by a special way of representing its elements, but is understood abstractly and corresponds to the finite group . ${\displaystyle C_{4}}$ ${\displaystyle V}$ ${\displaystyle C_{2}\times C_{2}}$

Link panel

The Klein group of four operates on a carrier set of cardinality 4 and has four elements, e.g. B. , of which the neutral element is. Their (internal) connection of two elements results in one of the four elements - if the order of the respectively connected pairs is reversed, the same result ( commutative law ), in the case of (two-digit) connection of an element with itself the neutral element - and is shown by the following connection table specified: ${\displaystyle 1,a,b,ab}$ ${\displaystyle 1}$

Linking table of the Klein group of four in color		For comparison, the link table for cyclic group C 4

${\displaystyle \circ }$	1	a	b	from
1	1	a	b	from
a	a	1	from	b
b	b	from	1	a
from	from	b	a	1

The graphics on the right show the link table of the Klein group of four and that of the cyclic group of the same order in color. The colored linking table of the Klein group of four follows the order of the elements in the table on the left. The neutral element is black in each case. Colored link tables, as in the graphics, are used in the MathWorld online encyclopedia for mathematics , as are those in grayscale. ${\displaystyle C_{4}}$

As with all commutative groups, this table of the two-digit connection is symmetrical with respect to the main diagonal, which in the group of four - unlike z. B. with the cyclic group  - is occupied solely by the neutral element. Thus every element is also (on both sides) the inverse element of itself; so each element is involutive . ${\displaystyle \circ }$ ${\displaystyle V}$ ${\displaystyle C_{4}}$

The copies of the header and input line, with the usual notation as here in the 1st line or the 1st column, identify the neutral element (on both sides) , which is also called "identity" as an identical illustration of the elements. ${\displaystyle 1}$

properties

The Klein group of four is a commutative , but not a cyclic group. Its subgroups are {1}, {1, a}, {1, b}, {1, ab}, {1, a, b, ab} and all normal , so the group of four is not a finite simple group . The non-neutral elements have the element order 2, each element forms its own conjugation class . ${\displaystyle V}$ ${\displaystyle a,b,ab}$

The group of four corresponds to the (Abelian and non-cyclic) finite group - a direct product of two examples of the cyclic group , which is the smallest non-trivial group and the only one in group order 2. The abstract properties of the group of four can be shown using the example of different point groups and multiplicative groups that are isomorphic to it . ${\displaystyle C_{2}\times C_{2}}$ ${\displaystyle C_{2}}$

Occur

The group of four occurs, for example, as the symmetry group of a non- equiangular diamond or a non- equilateral rectangle (which are therefore not a square; its symmetry group would be the dihedral group (group order 8) and the rotating group of a square is an example of the cyclic group ): ${\displaystyle V}$ ${\displaystyle D_{4}}$ ${\displaystyle C_{4}}$

The four elements are: as the identity (or rotation by 0 °), as the reflection on the vertical central axis, as the reflection on the horizontal central axis, and as the 180 ° rotation around the center, which is also combined horizontal and vertical mirroring can be understood. With the corners of a rectangle labeled as above, the permutation representation delivers ${\displaystyle 1}$ ${\displaystyle a}$ ${\displaystyle b}$ ${\displaystyle ab}$

${\displaystyle \left(A,B,C,D\right)\mapsto \left(A,B,C,D\right)}$, The element representing ${\displaystyle 1}$

${\displaystyle \left(A,B,C,D\right)\mapsto \left(B,A,D,C\right)}$, The element representing ${\displaystyle a}$

${\displaystyle \left(A,B,C,D\right)\mapsto \left(D,C,B,A\right)}$, The element representing ${\displaystyle b}$

${\displaystyle \left(A,B,C,D\right)\mapsto \left(C,D,A,B\right)}$, The element representing ${\displaystyle ab}$

and with the notation of the permutations in cycle notation

${\displaystyle V=\{\mathbf {id} =(A)(B)(C)(D),(A,B)(C,D),(A,D)(B,C),(A,C)(B,D)\}}$

In this representation, the commutator group is a normal divisor of the alternating group and also a normal divisor of the symmetrical group . In Galois theory , the existence of Klein's group of four in this representation explains the existence of the solution formula for equations of the fourth degree . ${\displaystyle V}$ ${\displaystyle A_{4}}$ ${\displaystyle S_{4}}$

Furthermore, the group of four is isomorphic to

• ${\displaystyle \mathbb {Z} /2\mathbb {Z} \times \mathbb {Z} /2\mathbb {Z} }$,
• the dihedral group of order 4 ( ), ${\displaystyle D_{2}}$
• the unit group of the ring (these are the remainder classes of 1, 3, 5 and 7 under multiplication modulo 8), ${\displaystyle \mathbb {Z} /8\mathbb {Z} }$
• the unit group of the ring (these are the remainder classes of 1, 5, 7 and 11 under multiplication modulo 12), ${\displaystyle \mathbb {Z} /12\mathbb {Z} }$
• the automorphism group of the following graph:

• that of the involutions with any body and ${\displaystyle a,b\colon K^{*}\to K^{*}}$ ${\displaystyle K}$

${\displaystyle a:}$	${\displaystyle x\mapsto -x}$
${\displaystyle b:}$	${\displaystyle x\mapsto x^{-1}}$

created group with the sequential execution as a group link.

Generators and Relations

The Klein group of four is not a cyclic group . It is generated by any two of the three group elements with order 2 under consideration of certain relations, for example by and . You get the presentation like this: ${\displaystyle a}$ ${\displaystyle b}$

${\displaystyle V=\left\langle a,b\mid a^{2}=b^{2}={(ab)}^{2}=1\right\rangle }$.

Commutativity follows: . ${\displaystyle ba=(a^{-1}b^{-1})^{-1}=(ab)^{-1}=ab}$

Representations

Representation in ℝ²

The small group of four can be generated from 2 reflections on the coordinate axes. Group linking is matrix multiplication. The product of the two reflections is a rotation of 180 ° around the coordinate origin.

${\displaystyle V\cong \left\langle {\begin{pmatrix}-1&0\\0&1\end{pmatrix}},{\begin{pmatrix}1&0\\0&-1\end{pmatrix}}\right\rangle }$

Representation in ℝ³

The small group of four can be generated from two 180 ° rotations around the coordinate axes. Group linking is matrix multiplication. The product of the two rotations is a rotation of 180 ° around the third coordinate axis.

${\displaystyle V\cong \left\langle {\begin{pmatrix}1&0&0\\0&-1&0\\0&0&-1\end{pmatrix}},{\begin{pmatrix}-1&0&0\\0&1&0\\0&0&-1\end{pmatrix}}\right\rangle }$

The regular representation

The regular representation of (is set here ) over a field (e.g. ) is the following group homomorphism into the group of invertible 4 × 4 matrices . is the mapping matrix for the linear mapping that maps the basis of the 4-dimensional vector space , i.e. the 4 basic elements are understood as elements of the group of four and multiplied by. Then of course there is the 4 × 4 identity matrix . For the determination of note that the basis is mapped onto , that is to say onto , the representing matrix is ​​therefore ${\displaystyle V=\{e,a,b,c\}}$ ${\displaystyle c=ab}$ ${\displaystyle K}$ ${\displaystyle K=\mathbb {R} }$ ${\displaystyle \rho \colon V\to \mathrm {GL} _{4}(K)}$ ${\displaystyle \rho (x)}$ ${\displaystyle e,a,b,c}$ ${\displaystyle Ke+Ka+Kb+Kc}$ ${\displaystyle xe,xa,xb,xc}$ ${\displaystyle x}$ ${\displaystyle \rho (e)}$ ${\displaystyle \rho (a)}$ ${\displaystyle e,a,b,c}$ ${\displaystyle ae,aa,ab,ac}$ ${\displaystyle a,e,c,b}$

${\displaystyle \rho (a)={\begin{pmatrix}0&1&0&0\\1&0&0&0\\0&0&0&1\\0&0&1&0\end{pmatrix}}}$.

That's how you determine

${\displaystyle \rho (b)={\begin{pmatrix}0&0&1&0\\0&0&0&1\\1&0&0&0\\0&1&0&0\end{pmatrix}},\quad \quad \rho (c)={\begin{pmatrix}0&0&0&1\\0&0&1&0\\0&1&0&0\\1&0&0&0\end{pmatrix}}}$

thats why

${\displaystyle \left\{{\begin{pmatrix}1&0&0&0\\0&1&0&0\\0&0&1&0\\0&0&0&1\end{pmatrix}},{\begin{pmatrix}0&1&0&0\\1&0&0&0\\0&0&0&1\\0&0&1&0\end{pmatrix}},{\begin{pmatrix}0&0&1&0\\0&0&0&1\\1&0&0&0\\0&1&0&0\end{pmatrix}},{\begin{pmatrix}0&0&0&1\\0&0&1&0\\0&1&0&0\\1&0&0&0\end{pmatrix}}\right\}\,\subset \,\mathrm {GL} _{4}(K)}$

a 4-element group that is isomorphic to Klein's group of four, and the figure given is a group isomorphism. ${\displaystyle \rho }$

Irreducible representations

As a four-element Abelian group, Klein's group of four must have four irreducible representations . These are the following group homomorphisms : ${\displaystyle V=\{e,a,b,c\}}$ ${\displaystyle \sigma _{e},\sigma _{a},\sigma _{b},\sigma _{c}\colon V\to \{1,-1\}}$

${\displaystyle \sigma _{e}(x)=1}$  for all ${\displaystyle x\in V}$

${\displaystyle \sigma _{a}(x)={\begin{cases}1&{\text{ für }}x\in \{e,a\}\\-1&{\text{ sonst}}\end{cases}}}$

${\displaystyle \sigma _{b}(x)={\begin{cases}1&{\text{ für }}x\in \{e,b\}\\-1&{\text{ sonst}}\end{cases}}}$

${\displaystyle \sigma _{c}(x)={\begin{cases}1&{\text{ für }}x\in \{e,c\}\\-1&{\text{ sonst}}\end{cases}}}$

Note that these homomorphisms with respect to the point-wise multiplication of mappings again form a group and that , is a group homomorphism that is an isomorphism . This shows that it is isomorphic to its dual group . ${\displaystyle \sigma \colon V\to \operatorname {Abb} (V,\mathbb {C} ),\,x\mapsto \sigma _{x}}$ ${\displaystyle V\to \{\sigma _{e},\sigma _{a},\sigma _{b},\sigma _{c}\}}$ ${\displaystyle V}$

Automorphism group

An automorphism of Klein's group of four must leave the orders of the group elements fixed, i.e. it can at most permute the three elements of order 2 . In fact, every mapping that leaves and permutes is an automorphism. This is because the connection on can be described in such a way that the product of two identical elements of order 2 equals the neutral element and the product of two different elements of order 2 is the third element of order 2, and that remains obtained with permutations of the elements of order 2. Hence the automorphism group is from isomorphic to symmetric group S ₃ . ${\displaystyle V}$ ${\displaystyle a,b,c}$ ${\displaystyle e}$ ${\displaystyle a,b,c}$ ${\displaystyle V}$ ${\displaystyle V}$

Web links

• www.mathe.tu-freiberg.de/~hebisch/cafe/algebra: Small groups

Individual evidence

1. ↑ Felix Klein: Lectures on the icosahedron and the solution of the equations of the fifth degree . Teubner, Leipzig 1884, p. 27 f . (VIII, 260, online ).
2. ↑ MathWorld: Finite Group C_2 × C_2
3. ^ Kurt Meyberg: Algebra II. Carl Hanser Verlag (1976), ISBN 3-446-12172-2 , example 9.1.3 c ₁