# A4 (group) - A4 (Gruppe)

The ${\displaystyle A_{4}}$( 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 ${\displaystyle S_{4}}$, it is the ${\displaystyle A_{4}}$around the subgroup that consists of all even permutations . Geometrically, the${\displaystyle A_{4}}$as a group of rotations of the regular tetrahedron on itself.

## Geometrical introduction

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

If one looks at the rotations that transform a regular tetrahedron into itself, one finds 12 possibilities: [1]

• 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 midpoints of the edges 12 and 34 (1, 2, 3 and 4 denote tetrahedron 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 of 120 ° around the corner ${\displaystyle i}$running height, in the positive direction of rotation ( i.e. counterclockwise ) seen from the pierced corner.
• ${\displaystyle d_{i}^{2}}$ be the rotation by 240 ° around the corner ${\displaystyle i}$ running height, also with the direction of rotation specified above.

These rotations can be combined by executing them one after the other , which again results in a rotation from the list above. You just write two rotations (often without a link, or with${\displaystyle \cdot }$ or ${\displaystyle \circ }$) next to each other and means that first the right-hand and then the left-hand rotation must be performed. The spelling${\displaystyle d_{i}^{2}}$ Already makes clear that the rotation by 240 ° is equal to the double execution of the rotation by 120 °.

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

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 show some relationships better 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 shown in the graphic are used in the MathWorld online encyclopedia for mathematics , as are those in grayscale. [2]

## 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. Every element of the${\displaystyle A_{4}}$ can therefore be used as a permutation of the set ${\displaystyle \{1,2,3,4\}}$be understood. If you use the usual two-line form and the cycle notation , you get:

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

You can see at a glance that every element of the ${\displaystyle A_{4}}$can be written as a product of an even number of transpositions (= two permutations). The associated permutations are also called straight , that is, the${\displaystyle A_{4}}$ consists exactly of the even permutations of the set ${\displaystyle \{1,2,3,4\}}$. With that the${\displaystyle A_{4}}$as the core of the Signum image:${\displaystyle S_{4}\rightarrow \{-1,1\}}$ on, where ${\displaystyle S_{4}}$is the fourth degree symmetric group .

## characteristics

### Subgroups

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

All subgroups of the ${\displaystyle A_{4}}$[3] are given in the adjacent drawing.

${\displaystyle V:=\{e,a,b,c\}}$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. The${\displaystyle A_{4}}$ is an example of this phenomenon because it has no subgroup of order 6.

### Normal divisor, solvability

The ${\displaystyle A_{4}}$is not Abelian , because

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

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

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

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

${\displaystyle V}$is the commutator group of${\displaystyle A_{4}}$, [4] especially a normal divisor and it holds${\displaystyle A_{4}/V\cong \mathbb {Z} /3\mathbb {Z} }$

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

### Semi-direct product

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

### Generators and relations

Groups can also be described by specifying a generating system 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: [5]

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

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

### Charactertafel

The character board of the${\displaystyle A_{4}}$looks like this: [6]

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

2. MathWorld: Tetrahedral Group On this website there are the link panels (in color) of the tetrahedral group ${\displaystyle T_{d}}$ and that of its subgroup, the tetrahedral rotating group ${\displaystyle T}$ that are isomorphic to the alternating group ${\displaystyle A_{4}}$is. The order of the elements for the color graphics is not specified there.