S3 (group)

In the mathematical sub-area of group theory, the symmetrical group denotes a specific group with 6 elements. It can be described as a group of the six permutations of a three-element set. Alternative names are and . It is isomorphic with the Dïedergruppe  , the group of congruence maps of the equilateral triangle on itself. ${\displaystyle S_{3}}$ ${\displaystyle {\mathfrak {S}}_{3}}$ ${\displaystyle \mathop {\mathrm {Sym} } \nolimits _{3}}$ ${\displaystyle D_{3}}$

introduction

The effects of the pictures , , , and ${\displaystyle d}$ ${\displaystyle d^{2}}$ ${\displaystyle s_{1}}$ ${\displaystyle s_{2}}$ ${\displaystyle s_{3}}$

If you look at the congruence maps that transform an equilateral triangle into themselves, you find 6 possibilities:

• the identical figure , ${\displaystyle e}$
• the rotation of 120 ° around the center of the triangle, ${\displaystyle d}$
• the rotation of 240 ° around the center of the triangle, ${\displaystyle d^{2}}$
• the three reflections and at the three perpendiculars of the triangle. ${\displaystyle s_{1},s_{2}}$ ${\displaystyle s_{3}}$

These congruence maps can be combined by executing them one after the other , whereby a congruence map is obtained again. One simply writes two congruence maps (often without linking symbols, or with or ) next to each other and means that ${\displaystyle \cdot }$ ${\displaystyle \circ }$

first the one on the right and then the one on the left

Congruence mapping is to be carried out. The notation makes it clear that the rotation by 240 ° is equal to the double execution of the rotation by 120 °. ${\displaystyle d^{2}}$

In this way the alternate-element group of all congruence maps of the equilateral triangle is obtained. If you enter all the links formed in this way in a link table , you get ${\displaystyle S_{3}=\left\{e,d,d^{2},s_{1},s_{2},s_{3}\right\}}$

Linking table of the symmetrical group S ₃ in color. The neutral element (the identical picture) is black

${\displaystyle \cdot }$	${\displaystyle e}$	${\displaystyle d}$	${\displaystyle d^{2}}$	${\displaystyle s_{1}}$	${\displaystyle s_{2}}$	${\displaystyle s_{3}}$
${\displaystyle e}$	${\displaystyle e}$	${\displaystyle d}$	${\displaystyle d^{2}}$	${\displaystyle s_{1}}$	${\displaystyle s_{2}}$	${\displaystyle s_{3}}$
${\displaystyle d}$	${\displaystyle d}$	${\displaystyle d^{2}}$	${\displaystyle e}$	${\displaystyle s_{3}}$	${\displaystyle s_{1}}$	${\displaystyle s_{2}}$
${\displaystyle d^{2}}$	${\displaystyle d^{2}}$	${\displaystyle e}$	${\displaystyle d}$	${\displaystyle s_{2}}$	${\displaystyle s_{3}}$	${\displaystyle s_{1}}$
${\displaystyle s_{1}}$	${\displaystyle s_{1}}$	${\displaystyle s_{2}}$	${\displaystyle s_{3}}$	${\displaystyle e}$	${\displaystyle d}$	${\displaystyle d^{2}}$
${\displaystyle s_{2}}$	${\displaystyle s_{2}}$	${\displaystyle s_{3}}$	${\displaystyle s_{1}}$	${\displaystyle d^{2}}$	${\displaystyle e}$	${\displaystyle d}$
${\displaystyle s_{3}}$	${\displaystyle s_{3}}$	${\displaystyle s_{1}}$	${\displaystyle s_{2}}$	${\displaystyle d}$	${\displaystyle d^{2}}$	${\displaystyle e}$

If you want to the product of two elements of figure out, so they were looking in the truth table first with marked column, with marked line on; at the intersection of this column and row is the product. ${\displaystyle ba}$ ${\displaystyle a,b}$ ${\displaystyle S_{3}}$ ${\displaystyle a}$ ${\displaystyle b}$

The graphic on the right shows the link table in color. This colored link table follows the order of the elements in the table on the left. Colored link tables, as in the graphic, are used in the online encyclopedia for mathematics MathWorld , as are those in grayscale.

If one generalizes this construction by replacing the equilateral triangle with a regular corner, one arrives at the concept of the dihedral group . Therefore, the group discussed here is also referred to as. ${\displaystyle n}$ ${\displaystyle S_{3}}$ ${\displaystyle D_{3}}$

Elements of the S ₃ as permutations

A congruence mapping of the equilateral triangle is already clearly defined by how the corners labeled 1, 2 and 3 are mapped onto one another. Each element of can therefore be understood as a permutation of the set . In the following, the two-line form is given first, followed by the cycle notation of the elements and their order : ${\displaystyle S_{3}}$ ${\displaystyle \{1,2,3\}}$

${\displaystyle {\begin{array}{rcccll}e&=&{\begin{pmatrix}1&2&3\\1&2&3\end{pmatrix}}&=&(1)&\qquad \mathrm {ord} \left(e\right)=1\\\\d&=&{\begin{pmatrix}1&2&3\\2&3&1\end{pmatrix}}&=&(1~2~3)&\qquad \mathrm {ord} \left(d\right)=3\\\\d^{2}&=&{\begin{pmatrix}1&2&3\\3&1&2\end{pmatrix}}&=&(1~3~2)&\qquad \mathrm {ord} \left(d^{2}\right)=3\\\\s_{1}&=&{\begin{pmatrix}1&2&3\\1&3&2\end{pmatrix}}&=&(2~3)&\qquad \mathrm {ord} \left(s_{1}\right)=2\\\\s_{2}&=&{\begin{pmatrix}1&2&3\\3&2&1\end{pmatrix}}&=&(1~3)&\qquad \mathrm {ord} \left(s_{2}\right)=2\\\\s_{3}&=&{\begin{pmatrix}1&2&3\\2&1&3\end{pmatrix}}&=&(1~2)&\qquad \mathrm {ord} \left(s_{3}\right)=2\end{array}}}$

properties

Not an Abelian group

The group is not an Abelian group , as can be seen from the table above (it is not symmetrical to the main diagonal); for example . Except for isomorphism, it is the smallest non-Abelian group, that is, every non-Abelian group is either isomorphic to or has more elements. ${\displaystyle S_{3}}$ ${\displaystyle s_{2}s_{1}=d\neq d^{2}=s_{1}s_{2}}$ ${\displaystyle S_{3}}$

Subgroups and normal divisors

The subgroups next to the trivial subgroups and themselves are: ${\displaystyle \{e\}}$ ${\displaystyle S_{3}}$

• ${\displaystyle A_{3}:=\left\{e,d,d^{2}\right\}\cong \mathbb {Z} /3\mathbb {Z} }$. This subgroup (the group of rotations) is a normal divisor and is also known as an alternating group of degree 3.
• ${\displaystyle \{e,s_{1}\}\cong \{e,s_{2}\}\cong \{e,s_{3}\}\cong \mathbb {Z} /2\mathbb {Z} }$. These subgroups (the groups of reflections) are not normal subgroups; for example is . ${\displaystyle d\{e,s_{1}\}d^{-1}\,=\,d\{e,s_{1}\}d^{2}\,=\,\{e,s_{2}\}}$
• The center of is trivial (consists only of ). Thus, an element that is different from one another commutes only with powers of itself. ${\displaystyle S_{3}}$ ${\displaystyle \{e\}}$ ${\displaystyle e}$

Generators and Relations

Groups can also be described by specifying a system of generators and relations that the generators must fulfill. Generators and relations are noted, separated by the symbol     , in angle brackets. The group is then the free group generated by the generators modulo the normal divisor generated by the relations. In this sense: ${\displaystyle \mid }$

${\displaystyle S_{3}=\langle d,s\mid d^{3},s^{2},dsds\rangle }$

Irreducible representations

Except for equivalence, it has three irreducible representations , two one-dimensional and one two-dimensional. To specify these representations, it is sufficient to specify the images of and , because these elements create the group. ${\displaystyle S_{3}}$ ${\displaystyle d}$ ${\displaystyle s_{1}}$

• The trivial representation: ${\displaystyle S_{3}\rightarrow \mathbb {C} :\,\,d\mapsto 1,s_{1}\mapsto 1}$
• The Signum illustration: ${\displaystyle S_{3}\rightarrow \mathbb {C} :\,\,d\mapsto 1,s_{1}\mapsto -1}$
• The two-dimensional representation: . ${\displaystyle S_{3}\rightarrow M_{2}(\mathbb {C} ):\,\,d\mapsto {\begin{bmatrix}e^{2\pi i/3}&0\\0&e^{-2\pi i/3}\end{bmatrix}},s_{1}\mapsto {\begin{bmatrix}0&1\\1&0\end{bmatrix}}}$

While you get a different two-dimensional representation when by replaced, but this is equivalent to the specified. These considerations lead to the following character table : ${\displaystyle s_{1}}$ ${\displaystyle s_{2}}$

${\displaystyle S_{3}}$	${\displaystyle 1}$	${\displaystyle 3}$	${\displaystyle 2}$
	${\displaystyle 1}$	${\displaystyle (1,2)}$	${\displaystyle (1,2,3)}$
${\displaystyle \chi _{1}}$	${\displaystyle 1}$	${\displaystyle 1}$	${\displaystyle 1}$
${\displaystyle \chi _{2}}$	${\displaystyle 1}$	${\displaystyle -1}$	${\displaystyle 1}$
${\displaystyle \chi _{3}}$	${\displaystyle 2}$	${\displaystyle 0}$	${\displaystyle -1}$

Further examples

General linear group over ${\displaystyle \mathbb {Z} /2}$

The general linear group 2 of degree over the residue field , is isomorphic to . ${\displaystyle \mathbb {Z} /2=\mathbb {F} _{2}=\{0,1\}}$ ${\displaystyle GL(2,\mathbb {F} _{2})=\left\{{\begin{bmatrix}1&0\\0&1\end{bmatrix}},{\begin{bmatrix}1&1\\0&1\end{bmatrix}},{\begin{bmatrix}1&0\\1&1\end{bmatrix}},{\begin{bmatrix}0&1\\1&0\end{bmatrix}},{\begin{bmatrix}1&1\\1&0\end{bmatrix}},{\begin{bmatrix}0&1\\1&1\end{bmatrix}}\right\}}$ ${\displaystyle S_{3}}$

Transformation group

The fractional linear functions with coefficients from any field and the assignments ${\displaystyle s_{1},s_{2}}$ ${\displaystyle K}$

${\displaystyle s_{1}:}$	${\displaystyle X\mapsto 1-X}$
${\displaystyle s_{2}:}$	${\displaystyle X\mapsto X^{-1}}$

create a group that is isomorphic to the sequential execution as a group link . The other 4 group members are: ${\displaystyle G}$ ${\displaystyle S_{3}}$

${\displaystyle d:=s_{1}\circ s_{2}:}$	${\displaystyle X\mapsto {\tfrac {X-1}{X}}}$
${\displaystyle s_{3}:=d\circ s_{1}=s_{2}\circ d:}$	${\displaystyle X\mapsto {\tfrac {X}{X-1}}}$
${\displaystyle d^{2}:=d\circ d=s_{2}\circ s_{1}:}$	${\displaystyle X\mapsto {\tfrac {1}{1-X}}}$
${\displaystyle d^{3}:=d\circ d^{2}=e:}$	${\displaystyle X\mapsto X}$

The link panel is as above .
The 6 group members differ in the use of elements ${\displaystyle s\in G}$ ${\displaystyle x\in K\!\setminus \!\{0,1\}}$

${\displaystyle s_{K}:x\mapsto s_{K}(x):=s(x)}$

also in the value tables if has at least 5 elements. ${\displaystyle K}$

Automorphism group

This is isomorphic to the automorphism group of Klein's group of four . This follows easily from the observation that every permutation of the three elements of order 2 of the Klein group of four defines an automorphism. ${\displaystyle S_{3}}$

See also

• List of small groups

Web links

• Applet from the Technical University of Munich for the visualization of ${\displaystyle S_{3}}$

Individual evidence

1. ↑ Arno Mitschka: Elements of group theory. Study books mathematics, 1975, ISBN 3-451-16528-7 , section II.5
2. ↑ ^{a b} This order comes from the operator perspective, as it prevails with the sequential connection of mappings (also with the permutations). The sequence is irrelevant for pure group theory.
3. ↑ MathWorld: Symmetric Group The symmetric group S ₃ is not included, but S ₄ for example.
4. ^ K. Meyberg: Algebra, Part I. Carl Hanser Verlag, 1980, ISBN 3-446-13079-9 , example 2.4.2.c
5. ^ K. Meyberg: Algebra, Part I. Carl Hanser Verlag, 1980, ISBN 3-446-13079-9 , Example 2.7.18.c
6. ↑ JP Serre: Representations of finite groups. Vieweg, 1972, ISBN 3-528-03556-0 , §5.3
7. ^ Kurt Meyberg: Algebra II. Carl Hanser Verlag, 1976, ISBN 3-446-12172-2 , Example 9.7.1 b
8. ↑ If the body of the complex numbers, more precisely: the Riemann number ball , then it is a Möbius transformation . ${\displaystyle K}$