A ₄ (group)

The ( alternating group of 4th degree) is a specific 12-element group that is examined in the mathematical sub-area of group theory . It is closely related to the symmetrical group ; it is the subgroup that consists of all even permutations . Geometrically, it arises as a group of rotations of the regular tetrahedron on itself. ${\displaystyle A_{4}}$ ${\displaystyle S_{4}}$ ${\displaystyle A_{4}}$ ${\displaystyle A_{4}}$

Geometrical introduction

The twists and the tetrahedron ${\displaystyle c}$ ${\displaystyle d_{4}}$

If one considers the rotations that transform a regular tetrahedron into itself, one finds 12 possibilities:

• the identity , ${\displaystyle e}$
• three rotations of 180 ° around axes that run through the centers of two opposite edges,
• four rotations of 120 ° around the heights of the tetrahedron,
• four rotations of 240 ° around the heights of the tetrahedron.

Reflections are not considered here. We choose the following terms for the rotations:

• ${\displaystyle a}$ is the rotation by 180 ° around the straight line that runs through the center points of the edges 12 and 34 (1, 2, 3 and 4 denote tetrahedral corners as in the adjacent drawing).
• ${\displaystyle b}$ is the rotation by 180 ° around the straight line that runs through the midpoints of edges 13 and 24.
• ${\displaystyle c}$ is the rotation through 180 ° around the straight line that runs through the midpoints of the edges 14 and 23.
• ${\displaystyle d_{i}}$be the rotation by 120 ° around the height running through the corner , namely in the positive direction of rotation (that is, counterclockwise ) seen from the pierced corner. ${\displaystyle i}$
• ${\displaystyle d_{i}^{2}}$be the rotation by 240 ° around the height running through the corner , also with the direction of rotation given above. ${\displaystyle i}$

These rotations can be combined by executing them one after the other , which again results in a rotation from the list above. You simply write two rotations (often without a link, or with or ) side by side, meaning that the rotation on the right must be carried out first and then the rotation on the left. The notation makes it clear that the rotation by 240 ° is equal to the double execution of the rotation by 120 °. ${\displaystyle \cdot }$ ${\displaystyle \circ }$ ${\displaystyle d_{i}^{2}}$

In this way you get the 12-element group of all rotations of the regular tetrahedron on itself. ${\displaystyle A_{4}=\left\{e,a,b,c,d_{1},d_{1}^{2},d_{2},d_{2}^{2},d_{3},d_{3}^{2},d_{4},d_{4}^{2}\right\}}$

If you enter all the links formed in this way in a link table , you get

${\displaystyle \,\cdot }$ ${\displaystyle \,e}$ ${\displaystyle \,a}$ ${\displaystyle \,b}$ ${\displaystyle \,c}$ ${\displaystyle \,d_{1}}$ ${\displaystyle \,d_{1}^{2}}$ ${\displaystyle \,d_{2}}$ ${\displaystyle \,d_{2}^{2}}$ ${\displaystyle \,d_{3}}$ ${\displaystyle \,d_{3}^{2}}$ ${\displaystyle \,d_{4}}$ ${\displaystyle \,d_{4}^{2}}$ ${\displaystyle \,e}$ ${\displaystyle \,e}$ ${\displaystyle \,a}$ ${\displaystyle \,b}$ ${\displaystyle \,c}$ ${\displaystyle \,d_{1}}$ ${\displaystyle \,d_{1}^{2}}$ ${\displaystyle \,d_{2}}$ ${\displaystyle \,d_{2}^{2}}$ ${\displaystyle \,d_{3}}$ ${\displaystyle \,d_{3}^{2}}$ ${\displaystyle \,d_{4}}$ ${\displaystyle \,d_{4}^{2}}$ ${\displaystyle \,a}$ ${\displaystyle \,a}$ ${\displaystyle \,e}$ ${\displaystyle \,c}$ ${\displaystyle \,b}$ ${\displaystyle \,d_{4}}$ ${\displaystyle \,d_{3}^{2}}$ ${\displaystyle \,d_{3}}$ ${\displaystyle \,d_{4}^{2}}$ ${\displaystyle \,d_{2}}$ ${\displaystyle \,d_{1}^{2}}$ ${\displaystyle \,d_{1}}$ ${\displaystyle \,d_{2}^{2}}$ ${\displaystyle \,b}$ ${\displaystyle \,b}$ ${\displaystyle \,c}$ ${\displaystyle \,e}$ ${\displaystyle \,a}$ ${\displaystyle \,d_{2}}$ ${\displaystyle \,d_{4}^{2}}$ ${\displaystyle \,d_{1}}$ ${\displaystyle \,d_{3}^{2}}$ ${\displaystyle \,d_{4}}$ ${\displaystyle \,d_{2}^{2}}$ ${\displaystyle \,d_{3}}$ ${\displaystyle \,d_{1}^{2}}$ ${\displaystyle \,c}$ ${\displaystyle \,c}$ ${\displaystyle \,b}$ ${\displaystyle \,a}$ ${\displaystyle \,e}$ ${\displaystyle \,d_{3}}$ ${\displaystyle \,d_{2}^{2}}$ ${\displaystyle \,d_{4}}$ ${\displaystyle \,d_{1}^{2}}$ ${\displaystyle \,d_{1}}$ ${\displaystyle \,d_{4}^{2}}$ ${\displaystyle \,d_{2}}$ ${\displaystyle \,d_{3}^{2}}$ ${\displaystyle \,d_{1}}$ ${\displaystyle \,d_{1}}$ ${\displaystyle \,d_{3}}$ ${\displaystyle \,d_{4}}$ ${\displaystyle \,d_{2}}$ ${\displaystyle \,d_{1}^{2}}$ ${\displaystyle \,e}$ ${\displaystyle \,d_{3}^{2}}$ ${\displaystyle \,b}$ ${\displaystyle \,d_{4}^{2}}$ ${\displaystyle \,c}$ ${\displaystyle \,d_{2}^{2}}$ ${\displaystyle \,a}$ ${\displaystyle \,d_{1}^{2}}$ ${\displaystyle \,d_{1}^{2}}$ ${\displaystyle \,d_{4}^{2}}$ ${\displaystyle \,d_{2}^{2}}$ ${\displaystyle \,d_{3}^{2}}$ ${\displaystyle \,e}$ ${\displaystyle \,d_{1}}$ ${\displaystyle \,c}$ ${\displaystyle \,d_{4}}$ ${\displaystyle \,a}$ ${\displaystyle \,d_{2}}$ ${\displaystyle \,b}$ ${\displaystyle \,d_{3}}$ ${\displaystyle \,d_{2}}$ ${\displaystyle \,d_{2}}$ ${\displaystyle \,d_{4}}$ ${\displaystyle \,d_{3}}$ ${\displaystyle \,d_{1}}$ ${\displaystyle \,d_{4}^{2}}$ ${\displaystyle \,b}$ ${\displaystyle \,d_{2}^{2}}$ ${\displaystyle \,e}$ ${\displaystyle \,d_{1}^{2}}$ ${\displaystyle \,a}$ ${\displaystyle \,d_{3}^{2}}$ ${\displaystyle \,c}$ ${\displaystyle \,d_{2}^{2}}$ ${\displaystyle \,d_{2}^{2}}$ ${\displaystyle \,d_{3}^{2}}$ ${\displaystyle \,d_{1}^{2}}$ ${\displaystyle \,d_{4}^{2}}$ ${\displaystyle \,c}$ ${\displaystyle \,d_{3}}$ ${\displaystyle \,e}$ ${\displaystyle \,d_{2}}$ ${\displaystyle \,b}$ ${\displaystyle \,d_{4}}$ ${\displaystyle \,a}$ ${\displaystyle \,d_{1}}$ ${\displaystyle \,d_{3}}$ ${\displaystyle \,d_{3}}$ ${\displaystyle \,d_{1}}$ ${\displaystyle \,d_{2}}$ ${\displaystyle \,d_{4}}$ ${\displaystyle \,d_{2}^{2}}$ ${\displaystyle \,c}$ ${\displaystyle \,d_{4}^{2}}$ ${\displaystyle \,a}$ ${\displaystyle \,d_{3}^{2}}$ ${\displaystyle \,e}$ ${\displaystyle \,d_{1}^{2}}$ ${\displaystyle \,b}$ ${\displaystyle \,d_{3}^{2}}$ ${\displaystyle \,d_{3}^{2}}$ ${\displaystyle \,d_{2}^{2}}$ ${\displaystyle \,d_{4}^{2}}$ ${\displaystyle \,d_{1}^{2}}$ ${\displaystyle \,a}$ ${\displaystyle \,d_{4}}$ ${\displaystyle \,b}$ ${\displaystyle \,d_{1}}$ ${\displaystyle \,e}$ ${\displaystyle \,d_{3}}$ ${\displaystyle \,c}$ ${\displaystyle \,d_{2}}$ ${\displaystyle \,d_{4}}$ ${\displaystyle \,d_{4}}$ ${\displaystyle \,d_{2}}$ ${\displaystyle \,d_{1}}$ ${\displaystyle \,d_{3}}$ ${\displaystyle \,d_{3}^{2}}$ ${\displaystyle \,a}$ ${\displaystyle \,d_{1}^{2}}$ ${\displaystyle \,c}$ ${\displaystyle \,d_{2}^{2}}$ ${\displaystyle \,b}$ ${\displaystyle \,d_{4}^{2}}$ ${\displaystyle \,e}$ ${\displaystyle \,d_{4}^{2}}$ ${\displaystyle \,d_{4}^{2}}$ ${\displaystyle \,d_{1}^{2}}$ ${\displaystyle \,d_{3}^{2}}$ ${\displaystyle \,d_{2}^{2}}$ ${\displaystyle \,b}$ ${\displaystyle \,d_{2}}$ ${\displaystyle \,a}$ ${\displaystyle \,d_{3}}$ ${\displaystyle \,c}$ ${\displaystyle \,d_{1}}$ ${\displaystyle \,e}$ ${\displaystyle \,d_{4}}$

Linking table of the alternating group A 4 in color. The neutral element is black

The graphic on the right shows the link table in color. Such graphics make some connections easier to see than is the case with the use of numbers, letters or symbols. It should be noted that, in general, no particular arrangement can be identified for the elements of a group. A fixed rule, however, is that the neutral element is the first element of every row and column (top left corner). 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.

Representation as a permutation group

The rotations described above are already determined by how the corners marked 1, 2, 3 and 4 are mapped onto one another. Each element of can therefore be understood as a permutation of the set . If you use the usual two-line form and the cycle notation , you get: ${\displaystyle A_{4}}$ ${\displaystyle \{1,2,3,4\}}$

${\displaystyle {\begin{array}{rccclc}e&=&{\begin{pmatrix}1&2&3&4\\1&2&3&4\end{pmatrix}}&=&(1)&\mathrm {ord} (e)=1\\\\a&=&{\begin{pmatrix}1&2&3&4\\2&1&4&3\end{pmatrix}}&=&(1~2)(3~4)&\mathrm {ord} (a)=2\\\\b&=&{\begin{pmatrix}1&2&3&4\\3&4&1&2\end{pmatrix}}&=&(1~3)(2~4)&\mathrm {ord} (b)=2\\\\c&=&{\begin{pmatrix}1&2&3&4\\4&3&2&1\end{pmatrix}}&=&(1~4)(2~3)&\mathrm {ord} (c)=2\\\\d_{1}&=&{\begin{pmatrix}1&2&3&4\\1&4&2&3\end{pmatrix}}&=&(2~4~3)=(2~4)(4~3)&\mathrm {ord} (d_{1})=3\\\\d_{1}^{2}&=&{\begin{pmatrix}1&2&3&4\\1&3&4&2\end{pmatrix}}&=&(2~3~4)=(2~3)(3~4)&\mathrm {ord} (d_{1}^{2})=3\\\\d_{2}&=&{\begin{pmatrix}1&2&3&4\\3&2&4&1\end{pmatrix}}&=&(1~3~4)=(1~3)(3~4)&\mathrm {ord} (d_{2})=3\\\\d_{2}^{2}&=&{\begin{pmatrix}1&2&3&4\\4&2&1&3\end{pmatrix}}&=&(1~4~3)=(1~4)(4~3)&\mathrm {ord} (d_{2}^{2})=3\\\\d_{3}&=&{\begin{pmatrix}1&2&3&4\\4&1&3&2\end{pmatrix}}&=&(1~4~2)=(1~4)(4~2)&\mathrm {ord} (d_{3})=3\\\\d_{3}^{2}&=&{\begin{pmatrix}1&2&3&4\\2&4&3&1\end{pmatrix}}&=&(1~2~4)=(1~2)(2~4)&\mathrm {ord} (d_{3}^{2})=3\\\\d_{4}&=&{\begin{pmatrix}1&2&3&4\\2&3&1&4\end{pmatrix}}&=&(1~2~3)=(1~2)(2~3)&\mathrm {ord} (d_{4})=3\\\\d_{4}^{2}&=&{\begin{pmatrix}1&2&3&4\\3&1&2&4\end{pmatrix}}&=&(1~3~2)=(1~3)(3~2)&\mathrm {ord} (d_{4}^{2})=3\end{array}}}$

You can see at a glance that each element of can be written as a product of an even number of transpositions (= two-way permutations). The associated permutations are also called straight , that is, they consist exactly of the even permutations of the set . This is what appears as the core of the Signum figure: where the symmetrical group is of the fourth degree. ${\displaystyle A_{4}}$ ${\displaystyle A_{4}}$ ${\displaystyle \{1,2,3,4\}}$ ${\displaystyle A_{4}}$ ${\displaystyle S_{4}\rightarrow \{-1,1\}}$ ${\displaystyle S_{4}}$

properties

Subgroups

The subgroups of the ${\displaystyle A_{4}}$

All subgroups of are indicated in the adjacent drawing. ${\displaystyle A_{4}}$

${\displaystyle V:=\{e,a,b,c\}}$is isomorphic to Klein's group of four . According to Lagrange's theorem , the order of each subgroup divides the group order , in this case 12. Conversely, however, there need not be a subgroup of this order for every divisor of the group order. This is an example of this phenomenon because it has no subgroup of the 6th order. ${\displaystyle A_{4}}$

Normal divisor, solvability

It 's not Abelian , because ${\displaystyle A_{4}}$

${\displaystyle d_{1}\cdot a\neq a\cdot d_{1}}$

${\displaystyle A_{4}}$but is resolvable , like the series

${\displaystyle \{e\}\vartriangleleft \{e,a\}\vartriangleleft V\vartriangleleft A_{4}}$

shows. The sign means “ is normal divisor in” . ${\displaystyle \vartriangleleft }$

${\displaystyle V}$is the commutator group of , in particular a normal divisor and it applies ${\displaystyle A_{4}}$ ${\displaystyle A_{4}/V\cong \mathbb {Z} /3\mathbb {Z} }$

The two- and three-element subgroups are not normal divisors.

Semi-direct product

Since and have coprime group orders, it follows from the Schur-Zassenhaus theorem that the isomorphic to the semi-direct product , whereby the remainder class maps to the automorphism . ${\displaystyle \left\{e,d_{1},d_{1}^{2}\right\}\cong \mathbb {Z} /3\mathbb {Z} }$ ${\displaystyle V}$ ${\displaystyle A_{4}}$ ${\displaystyle V\times _{\theta }\mathbb {Z} /3\mathbb {Z} }$ ${\displaystyle \theta :\mathbb {Z} /3\mathbb {Z} \rightarrow \mathrm {Aut} (V)}$ ${\displaystyle {\overline {1}}\in \mathbb {Z} /3\mathbb {Z} }$ ${\displaystyle V\rightarrow V,\,x\mapsto d_{1}xd_{1}^{-1}}$

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 with the sign | separated, 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:

${\displaystyle A_{4}=\left\langle \alpha ,\beta \mid \alpha ^{3},\beta ^{3},\left(\alpha \beta \right)^{2}\right\rangle }$

It is easy to see that and satisfy the relations and that and generate the entire group, which is not yet sufficient for the proof. ${\displaystyle \alpha =d_{1}}$ ${\displaystyle \beta =d_{2}^{2}}$ ${\displaystyle d_{1}}$ ${\displaystyle d_{2}^{2}}$

Character board

The character table of looks like this: ${\displaystyle A_{4}}$

${\displaystyle A_{4}}$	${\displaystyle 1}$	${\displaystyle 3}$	${\displaystyle 4}$	${\displaystyle 4}$
	${\displaystyle 1}$	${\displaystyle (1,2)\,(3,4)}$	${\displaystyle (1,2,3)}$	${\displaystyle (1,3,2)}$
${\displaystyle \chi _{1}}$	${\displaystyle 1}$	${\displaystyle 1}$	${\displaystyle 1}$	${\displaystyle 1}$
${\displaystyle \chi _{2}}$	${\displaystyle 1}$	${\displaystyle 1}$	${\displaystyle \textstyle e^{\frac {2\pi i}{3}}}$	${\displaystyle \textstyle e^{\frac {4\pi i}{3}}}$
${\displaystyle \chi _{3}}$	${\displaystyle 1}$	${\displaystyle 1}$	${\displaystyle \textstyle e^{\frac {4\pi i}{3}}}$	${\displaystyle \textstyle e^{\frac {2\pi i}{3}}}$
${\displaystyle \chi _{4}}$	${\displaystyle 3}$	${\displaystyle -1}$	${\displaystyle 0}$	${\displaystyle 0}$

See also

• A5 (group)
• List of small groups
• Octahedral group

Individual evidence

1. ↑ Arno Mitschka: Elements of group theory , study books mathematics (1975), ISBN 3-451-16528-7 , section X, solution to IV.7
2. ↑ MathWorld: Tetrahedral Group This website contains the linking tables (in color) of the tetrahedral group and that of its subgroup, the tetrahedral rotating group , which is isomorphic to the alternating group . The order of the elements chosen for the color graphics is not specified there. ${\displaystyle T_{d}}$ ${\displaystyle T}$ ${\displaystyle A_{4}}$
3. ^ PJ Pahl, R. Damrath: Mathematical foundations of engineering informatics , Springer-Verlag (2000), ISBN 3-540-60501-0 , section 7.8.3. example 1
4. ^ K. Meyberg: Algebra, Part I , Carl Hanser Verlag (1980), ISBN 3-446-13079-9 , example 2.6.4
5. ^ K. Lamotke: Regular Solids and Isolated Singularities , Vieweg-Verlag (1986), ISBN 3-528-08958-X , Chapter I §8: Generators and Relations for the Finite Subgroups of SO (3)
6. ^ Kurt Meyberg: Algebra II. Carl Hanser Verlag (1976), ISBN 3-446-12172-2 , example 9.7.1 c