Affine group

The affine group or general affine group is a term from the mathematical branch of group theory . It is the group of all invertible , affine images of an affine space over a body in itself. ${\displaystyle K}$

definition

A bijective , affine mapping on a vector space has the form ${\displaystyle V}$

${\displaystyle f\colon V\rightarrow V,\,f(x)=Ax+v}$,

where is a vector space isomorphism , that is, an element of the general linear group , and is a fixed vector. That is, is the combination of a vector space isomorphism and a translation . To indicate the dependence on and , we also write . Composition and inversion of bijective affine mappings are again bijective and affine, because obviously holds ${\displaystyle A\in \mathrm {GL} (V)}$ ${\displaystyle v\in V}$ ${\displaystyle f}$ ${\displaystyle A}$ ${\displaystyle v}$ ${\displaystyle f=f_{A,v}}$

${\displaystyle (f_{A,v}\circ f_{B,w})(x)=f_{A,v}(f_{B,w}(x))=f_{A,v}(Bx+w)=ABx+Aw+v=f_{AB,Aw+v}(x)}$   so   ${\displaystyle f_{A,v}\circ f_{B,w}=f_{AB,Aw+v}}$

${\displaystyle f_{A,v}(x)=y\Leftrightarrow Ax+v=y\Leftrightarrow x=A^{-1}y-A^{-1}v}$   so   ${\displaystyle f_{A,v}^{-1}=f_{A^{-1},-A^{-1}v}}$

The bijective, affine maps therefore form a group, the so-called affine group or general affine group. Typical names are , or . If the n- dimensional vector space is over a body , one also writes . If further is finite with elements, then one simply designates with , because a finite body is uniquely determined by the number of its elements except for isomorphism. ${\displaystyle \mathrm {AGL} (V)}$ ${\displaystyle \mathrm {Aff} (V)}$ ${\displaystyle \mathrm {GA} (V)}$ ${\displaystyle V=K^{n}}$ ${\displaystyle K}$ ${\displaystyle \mathrm {AGL} _{n}(K)}$ ${\displaystyle K}$ ${\displaystyle q}$ ${\displaystyle \mathrm {AGL} _{n}(K)}$ ${\displaystyle \mathrm {AGL} _{n}(q)}$

Examples

AGL ₁ (ℝ)

Let be the one-dimensional real vector space. A bijective, affine mapping is then nothing more than an equation of a straight line ${\displaystyle V=\mathbb {R} ^{1}}$

${\displaystyle f(x)=ax+v}$with and . ${\displaystyle a\in \mathbb {R} ^{*}=\mathbb {R} \setminus \{0\}}$ ${\displaystyle v\in \mathbb {R} }$

${\displaystyle \mathrm {AGL} _{1}(\mathbb {R} )}$is the group of all non-constant linear equations. Each element has the shape with . So can with be identified, and for the group operations applies to this identification ${\displaystyle f=f_{a,v}}$ ${\displaystyle a\in \mathbb {R} ^{*},v\in \mathbb {R} }$ ${\displaystyle \mathrm {AGL} _{1}(\mathbb {R} )}$ ${\displaystyle \mathbb {R} ^{*}\times \mathbb {R} }$

${\displaystyle (a,v)\cdot (b,w)=(ab,aw+v)}$

${\displaystyle \textstyle (a,v)^{-1}=({\frac {1}{a}},{\frac {-v}{a}})}$.

${\displaystyle (1,0)}$is the neutral element .

AGL ₁ (5)

If you replace the above example with the finite field , you get the finite group with 20 elements described below . ${\displaystyle \mathbb {R} }$ ${\displaystyle \mathbb {Z} _{5}=\{0,1,2,3,4\}}$

A bijective, affine mapping has the shape ${\displaystyle \mathbb {Z} _{5}\rightarrow \mathbb {Z} _{5}}$

${\displaystyle f(x)=ax+v}$with and . ${\displaystyle a\in \mathbb {Z} _{5}^{*}=\{1,2,3,4\}}$ ${\displaystyle v\in \mathbb {Z} _{5}}$

If one denotes such an element simply av, then is

${\displaystyle \mathrm {AGL} _{1}(5)=\{10,11,12,13,14,20,21,22,23,24,30,31,32,33,34,40,41,42,43,44\}}$

and you have the following link table:

${\displaystyle \,\cdot }$	${\displaystyle \,10}$	${\displaystyle \,11}$	${\displaystyle \,12}$	${\displaystyle \,13}$	${\displaystyle \,14}$	${\displaystyle \,40}$	${\displaystyle \,41}$	${\displaystyle \,42}$	${\displaystyle \,43}$	${\displaystyle \,44}$	${\displaystyle \,20}$	${\displaystyle \,21}$	${\displaystyle \,22}$	${\displaystyle \,23}$	${\displaystyle \,24}$	${\displaystyle \,30}$	${\displaystyle \,31}$	${\displaystyle \,32}$	${\displaystyle \,33}$	${\displaystyle \,34}$
${\displaystyle \,10}$	${\displaystyle \,10}$	${\displaystyle \,11}$	${\displaystyle \,12}$	${\displaystyle \,13}$	${\displaystyle \,14}$	${\displaystyle \,40}$	${\displaystyle \,41}$	${\displaystyle \,42}$	${\displaystyle \,43}$	${\displaystyle \,44}$	${\displaystyle \,20}$	${\displaystyle \,21}$	${\displaystyle \,22}$	${\displaystyle \,23}$	${\displaystyle \,24}$	${\displaystyle \,30}$	${\displaystyle \,31}$	${\displaystyle \,32}$	${\displaystyle \,33}$	${\displaystyle \,34}$
${\displaystyle \,11}$	${\displaystyle \,11}$	${\displaystyle \,12}$	${\displaystyle \,13}$	${\displaystyle \,14}$	${\displaystyle \,10}$	${\displaystyle \,41}$	${\displaystyle \,42}$	${\displaystyle \,43}$	${\displaystyle \,44}$	${\displaystyle \,40}$	${\displaystyle \,21}$	${\displaystyle \,22}$	${\displaystyle \,23}$	${\displaystyle \,24}$	${\displaystyle \,20}$	${\displaystyle \,31}$	${\displaystyle \,32}$	${\displaystyle \,33}$	${\displaystyle \,34}$	${\displaystyle \,30}$
${\displaystyle \,12}$	${\displaystyle \,12}$	${\displaystyle \,13}$	${\displaystyle \,14}$	${\displaystyle \,10}$	${\displaystyle \,11}$	${\displaystyle \,42}$	${\displaystyle \,43}$	${\displaystyle \,44}$	${\displaystyle \,40}$	${\displaystyle \,41}$	${\displaystyle \,22}$	${\displaystyle \,23}$	${\displaystyle \,24}$	${\displaystyle \,20}$	${\displaystyle \,21}$	${\displaystyle \,32}$	${\displaystyle \,33}$	${\displaystyle \,34}$	${\displaystyle \,30}$	${\displaystyle \,31}$
${\displaystyle \,13}$	${\displaystyle \,13}$	${\displaystyle \,14}$	${\displaystyle \,10}$	${\displaystyle \,11}$	${\displaystyle \,12}$	${\displaystyle \,43}$	${\displaystyle \,44}$	${\displaystyle \,40}$	${\displaystyle \,41}$	${\displaystyle \,42}$	${\displaystyle \,23}$	${\displaystyle \,24}$	${\displaystyle \,20}$	${\displaystyle \,21}$	${\displaystyle \,22}$	${\displaystyle \,33}$	${\displaystyle \,34}$	${\displaystyle \,30}$	${\displaystyle \,31}$	${\displaystyle \,32}$
${\displaystyle \,14}$	${\displaystyle \,14}$	${\displaystyle \,10}$	${\displaystyle \,11}$	${\displaystyle \,12}$	${\displaystyle \,13}$	${\displaystyle \,44}$	${\displaystyle \,40}$	${\displaystyle \,41}$	${\displaystyle \,42}$	${\displaystyle \,43}$	${\displaystyle \,24}$	${\displaystyle \,20}$	${\displaystyle \,21}$	${\displaystyle \,22}$	${\displaystyle \,23}$	${\displaystyle \,34}$	${\displaystyle \,30}$	${\displaystyle \,31}$	${\displaystyle \,32}$	${\displaystyle \,33}$
${\displaystyle \,40}$	${\displaystyle \,40}$	${\displaystyle \,44}$	${\displaystyle \,43}$	${\displaystyle \,42}$	${\displaystyle \,41}$	${\displaystyle \,10}$	${\displaystyle \,14}$	${\displaystyle \,13}$	${\displaystyle \,12}$	${\displaystyle \,11}$	${\displaystyle \,30}$	${\displaystyle \,34}$	${\displaystyle \,33}$	${\displaystyle \,32}$	${\displaystyle \,31}$	${\displaystyle \,20}$	${\displaystyle \,24}$	${\displaystyle \,23}$	${\displaystyle \,22}$	${\displaystyle \,21}$
${\displaystyle \,41}$	${\displaystyle \,41}$	${\displaystyle \,40}$	${\displaystyle \,44}$	${\displaystyle \,43}$	${\displaystyle \,42}$	${\displaystyle \,11}$	${\displaystyle \,10}$	${\displaystyle \,14}$	${\displaystyle \,13}$	${\displaystyle \,12}$	${\displaystyle \,31}$	${\displaystyle \,30}$	${\displaystyle \,34}$	${\displaystyle \,33}$	${\displaystyle \,32}$	${\displaystyle \,21}$	${\displaystyle \,20}$	${\displaystyle \,24}$	${\displaystyle \,23}$	${\displaystyle \,22}$
${\displaystyle \,42}$	${\displaystyle \,42}$	${\displaystyle \,41}$	${\displaystyle \,40}$	${\displaystyle \,44}$	${\displaystyle \,43}$	${\displaystyle \,12}$	${\displaystyle \,11}$	${\displaystyle \,10}$	${\displaystyle \,14}$	${\displaystyle \,13}$	${\displaystyle \,32}$	${\displaystyle \,31}$	${\displaystyle \,30}$	${\displaystyle \,34}$	${\displaystyle \,33}$	${\displaystyle \,22}$	${\displaystyle \,21}$	${\displaystyle \,20}$	${\displaystyle \,24}$	${\displaystyle \,23}$
${\displaystyle \,43}$	${\displaystyle \,43}$	${\displaystyle \,42}$	${\displaystyle \,41}$	${\displaystyle \,40}$	${\displaystyle \,44}$	${\displaystyle \,13}$	${\displaystyle \,12}$	${\displaystyle \,11}$	${\displaystyle \,10}$	${\displaystyle \,14}$	${\displaystyle \,33}$	${\displaystyle \,32}$	${\displaystyle \,31}$	${\displaystyle \,30}$	${\displaystyle \,34}$	${\displaystyle \,23}$	${\displaystyle \,22}$	${\displaystyle \,21}$	${\displaystyle \,20}$	${\displaystyle \,24}$
${\displaystyle \,44}$	${\displaystyle \,44}$	${\displaystyle \,43}$	${\displaystyle \,42}$	${\displaystyle \,41}$	${\displaystyle \,40}$	${\displaystyle \,14}$	${\displaystyle \,13}$	${\displaystyle \,12}$	${\displaystyle \,11}$	${\displaystyle \,10}$	${\displaystyle \,34}$	${\displaystyle \,33}$	${\displaystyle \,32}$	${\displaystyle \,31}$	${\displaystyle \,30}$	${\displaystyle \,24}$	${\displaystyle \,23}$	${\displaystyle \,22}$	${\displaystyle \,21}$	${\displaystyle \,20}$
${\displaystyle \,20}$	${\displaystyle \,20}$	${\displaystyle \,22}$	${\displaystyle \,24}$	${\displaystyle \,21}$	${\displaystyle \,23}$	${\displaystyle \,30}$	${\displaystyle \,32}$	${\displaystyle \,34}$	${\displaystyle \,31}$	${\displaystyle \,33}$	${\displaystyle \,40}$	${\displaystyle \,42}$	${\displaystyle \,44}$	${\displaystyle \,41}$	${\displaystyle \,43}$	${\displaystyle \,10}$	${\displaystyle \,12}$	${\displaystyle \,14}$	${\displaystyle \,11}$	${\displaystyle \,13}$
${\displaystyle \,21}$	${\displaystyle \,21}$	${\displaystyle \,23}$	${\displaystyle \,20}$	${\displaystyle \,22}$	${\displaystyle \,24}$	${\displaystyle \,31}$	${\displaystyle \,33}$	${\displaystyle \,30}$	${\displaystyle \,32}$	${\displaystyle \,34}$	${\displaystyle \,41}$	${\displaystyle \,43}$	${\displaystyle \,40}$	${\displaystyle \,42}$	${\displaystyle \,44}$	${\displaystyle \,11}$	${\displaystyle \,13}$	${\displaystyle \,10}$	${\displaystyle \,12}$	${\displaystyle \,14}$
${\displaystyle \,22}$	${\displaystyle \,22}$	${\displaystyle \,24}$	${\displaystyle \,21}$	${\displaystyle \,23}$	${\displaystyle \,20}$	${\displaystyle \,32}$	${\displaystyle \,34}$	${\displaystyle \,31}$	${\displaystyle \,33}$	${\displaystyle \,30}$	${\displaystyle \,42}$	${\displaystyle \,44}$	${\displaystyle \,41}$	${\displaystyle \,43}$	${\displaystyle \,40}$	${\displaystyle \,12}$	${\displaystyle \,14}$	${\displaystyle \,11}$	${\displaystyle \,13}$	${\displaystyle \,10}$
${\displaystyle \,23}$	${\displaystyle \,23}$	${\displaystyle \,20}$	${\displaystyle \,22}$	${\displaystyle \,24}$	${\displaystyle \,21}$	${\displaystyle \,33}$	${\displaystyle \,30}$	${\displaystyle \,32}$	${\displaystyle \,34}$	${\displaystyle \,31}$	${\displaystyle \,43}$	${\displaystyle \,40}$	${\displaystyle \,42}$	${\displaystyle \,44}$	${\displaystyle \,41}$	${\displaystyle \,13}$	${\displaystyle \,10}$	${\displaystyle \,12}$	${\displaystyle \,14}$	${\displaystyle \,11}$
${\displaystyle \,24}$	${\displaystyle \,24}$	${\displaystyle \,21}$	${\displaystyle \,23}$	${\displaystyle \,20}$	${\displaystyle \,22}$	${\displaystyle \,34}$	${\displaystyle \,31}$	${\displaystyle \,33}$	${\displaystyle \,30}$	${\displaystyle \,32}$	${\displaystyle \,44}$	${\displaystyle \,41}$	${\displaystyle \,43}$	${\displaystyle \,40}$	${\displaystyle \,42}$	${\displaystyle \,14}$	${\displaystyle \,11}$	${\displaystyle \,13}$	${\displaystyle \,10}$	${\displaystyle \,12}$
${\displaystyle \,30}$	${\displaystyle \,30}$	${\displaystyle \,33}$	${\displaystyle \,31}$	${\displaystyle \,34}$	${\displaystyle \,32}$	${\displaystyle \,20}$	${\displaystyle \,23}$	${\displaystyle \,21}$	${\displaystyle \,24}$	${\displaystyle \,22}$	${\displaystyle \,10}$	${\displaystyle \,13}$	${\displaystyle \,11}$	${\displaystyle \,14}$	${\displaystyle \,12}$	${\displaystyle \,40}$	${\displaystyle \,43}$	${\displaystyle \,41}$	${\displaystyle \,44}$	${\displaystyle \,42}$
${\displaystyle \,31}$	${\displaystyle \,31}$	${\displaystyle \,34}$	${\displaystyle \,32}$	${\displaystyle \,30}$	${\displaystyle \,33}$	${\displaystyle \,21}$	${\displaystyle \,24}$	${\displaystyle \,22}$	${\displaystyle \,20}$	${\displaystyle \,23}$	${\displaystyle \,11}$	${\displaystyle \,14}$	${\displaystyle \,12}$	${\displaystyle \,10}$	${\displaystyle \,13}$	${\displaystyle \,41}$	${\displaystyle \,44}$	${\displaystyle \,42}$	${\displaystyle \,40}$	${\displaystyle \,43}$
${\displaystyle \,32}$	${\displaystyle \,32}$	${\displaystyle \,30}$	${\displaystyle \,33}$	${\displaystyle \,31}$	${\displaystyle \,34}$	${\displaystyle \,22}$	${\displaystyle \,20}$	${\displaystyle \,23}$	${\displaystyle \,21}$	${\displaystyle \,24}$	${\displaystyle \,12}$	${\displaystyle \,10}$	${\displaystyle \,13}$	${\displaystyle \,11}$	${\displaystyle \,14}$	${\displaystyle \,42}$	${\displaystyle \,40}$	${\displaystyle \,43}$	${\displaystyle \,41}$	${\displaystyle \,44}$
${\displaystyle \,33}$	${\displaystyle \,33}$	${\displaystyle \,31}$	${\displaystyle \,34}$	${\displaystyle \,32}$	${\displaystyle \,30}$	${\displaystyle \,23}$	${\displaystyle \,21}$	${\displaystyle \,24}$	${\displaystyle \,22}$	${\displaystyle \,20}$	${\displaystyle \,13}$	${\displaystyle \,11}$	${\displaystyle \,14}$	${\displaystyle \,12}$	${\displaystyle \,10}$	${\displaystyle \,43}$	${\displaystyle \,41}$	${\displaystyle \,44}$	${\displaystyle \,42}$	${\displaystyle \,40}$
${\displaystyle \,34}$	${\displaystyle \,34}$	${\displaystyle \,32}$	${\displaystyle \,30}$	${\displaystyle \,33}$	${\displaystyle \,31}$	${\displaystyle \,24}$	${\displaystyle \,22}$	${\displaystyle \,20}$	${\displaystyle \,23}$	${\displaystyle \,21}$	${\displaystyle \,14}$	${\displaystyle \,12}$	${\displaystyle \,10}$	${\displaystyle \,13}$	${\displaystyle \,11}$	${\displaystyle \,44}$	${\displaystyle \,42}$	${\displaystyle \,40}$	${\displaystyle \,43}$	${\displaystyle \,41}$

The arrangement of the elements was chosen so that the upper left parts of the link table show the 5-element subgroup and the 10-element subgroup . These are isomorphic to or (see special affine group below ). The latter is obviously not Abelian. Since there are only two 10-element groups apart from isomorphism, see the list of small groups , it must be isomorphic to the dihedral group . ${\displaystyle \{10,11,12,13,14\}}$ ${\displaystyle \{10,11,12,13,14,40,41,42,43,44\}}$ ${\displaystyle \mathbb {Z} _{5}}$ ${\displaystyle \mathbb {Z} _{2}\rtimes \mathbb {Z} _{5}}$ ${\displaystyle D_{5}}$

The affine group as a semi-direct product

construction

We consider the affine general group over the vector space . The subgroup of translations on is isomorphic to the additive group and the subgroup operates as a group of automorphisms on the translations. Obviously every element from is a product of an element of the subgroup and the translation group . Hence, one has the following semi-direct product ${\displaystyle \mathrm {AGL} (V)}$ ${\displaystyle V}$ ${\displaystyle V}$ ${\displaystyle (V,+)}$ ${\displaystyle \mathrm {GL} (V)}$ ${\displaystyle \mathrm {AGL} (V)}$ ${\displaystyle \mathrm {GL} (V)}$ ${\displaystyle V}$

${\displaystyle \mathrm {AGL} (V)\cong V\rtimes \mathrm {GL} (V)}$.

For that means ${\displaystyle V=K^{n}}$

${\displaystyle \mathrm {AGL} _{n}(K)\cong K^{n}\rtimes \mathrm {GL} (n,K)}$.

Number of elements

This means that the order of the group above the body with elements can easily be traced back to the group order of : ${\displaystyle \mathrm {AGL} _{n}(q)}$ ${\displaystyle K}$ ${\displaystyle q=p^{k}}$ ${\displaystyle \mathrm {GL} _{n}(q)}$

${\displaystyle \mathrm {ord} (\mathrm {AGL} _{n}(q))=|K^{n}|\cdot \mathrm {ord} (\mathrm {GL} _{n}(q))=q^{n}\cdot \prod _{i=0}^{n-1}(q^{n}-q^{i})}$

Example AGL ₁ (5)

The above example has elements according to the number formula above and can be used as ${\displaystyle \mathrm {AGL} _{1}(5)}$ ${\displaystyle 5^{1}\cdot (5^{1}-5^{0})=20}$

${\displaystyle \mathrm {AGL} _{1}(5)\cong \mathbb {Z} _{5}\rtimes \mathbb {Z} _{5}^{*}}$

to be written. There , you get ${\displaystyle \mathbb {Z} _{4}\cong \mathbb {Z} _{5}^{*}}$

${\displaystyle \mathrm {AGL} _{1}(5)\cong \mathbb {Z} _{5}\rtimes \mathbb {Z} _{4}}$.

However, one has to pay attention to how automorphisms operate on. With the identified identifications, the generating element is mapped to the multiplication by 2 on . That is what is meant by the semi-direct product . ${\displaystyle \mathbb {Z} _{4}}$ ${\displaystyle \mathbb {Z} _{5}}$ ${\displaystyle 1\in \mathbb {Z} _{4}}$ ${\displaystyle \mathbb {Z} _{5}}$ ${\displaystyle \mathbb {Z} _{5}\rtimes \mathbb {Z} _{4}}$

Example AGL ₂ (2)

${\displaystyle \mathrm {AGL} _{2}(2)}$is the group of affine mappings of the two-dimensional vector space, it permutes the four vectors of this vector space and is therefore isomorphic to a subgroup of the symmetric group . According to the above, however, it is also , that is, must therefore be isomorphic. ${\displaystyle \mathbb {Z} _{2}}$ ${\displaystyle S_{4}}$ ${\displaystyle \mathrm {ord} (\mathrm {AGL} _{2}(2))=2^{2}\cdot (2^{2}-2^{0})\cdot (2^{2}-2)=4\cdot 3\cdot 2=24}$ ${\displaystyle \mathrm {AGL} _{2}(2)}$ ${\displaystyle S_{4}}$

The affine group as a matrix group

The affine groups turn out to be subsets of general linear groups. It's easy to calculate that

${\displaystyle K^{n}\rtimes \mathrm {GL} _{n}(K)\rightarrow \mathrm {GL} _{n+1}(K),\quad (v,A)\mapsto {\begin{pmatrix}a_{11}&\ldots &a_{1n}&v_{1}\\\vdots &\ddots &\vdots &\vdots \\a_{n1}&\ldots &a_{nn}&v_{n}\\0&\ldots &0&1\end{pmatrix}}}$

is an injective homomorphism. The isomorphism therefore shows that isomorphic to the group of matrices ${\displaystyle \mathrm {AGL} _{n}(K)\cong K^{n}\rtimes \mathrm {GL} _{n}(K)}$ ${\displaystyle \mathrm {AGL} _{n}(K)}$

${\displaystyle \{(c_{i,j})_{i,j=1,\ldots ,n}|c_{i,j}\in K,c_{n+1,1}=\ldots =c_{n+1,n}=0,c_{n+1,n+1}=1\}}$

is. In short: is a subgroup of . ${\displaystyle \mathrm {AGL} _{n}(K)}$ ${\displaystyle \mathrm {GL} _{n+1}(K)}$

Other affine groups

The special affine group

One has the determinant mapping on the affine groups

${\displaystyle \det :\mathrm {AGL} _{n}(K)\cong K^{n}\rtimes \mathrm {GL} _{n}(K)\rightarrow K^{*},(v,A)\mapsto \det(A)}$,

which is a homomorphism into the unit group of the body. Alternatively, you can also use the above embedding and define the determinant mapping to as a restriction of the determinant mapping to . ${\displaystyle \mathrm {AGL} _{n}(K)\subset \mathrm {GL} _{n+1}(K)}$ ${\displaystyle \mathrm {AGL} _{n}(K)}$ ${\displaystyle \mathrm {GL} _{n+1}(K)}$

The core of this homomorphism, i.e. the set of all elements with determinant 1, is then a normal divisor in , which is called the special affine group in analogy to the special linear group and is denoted by. ${\displaystyle \mathrm {AGL} _{n}(K)}$ ${\displaystyle \mathrm {ASL} _{n}(K)}$

In the above example it is obvious . ${\displaystyle \mathrm {AGL} _{1}(5)}$ ${\displaystyle \mathrm {ASL} _{1}(5)=\{10,11,12,13,14\}}$

More generally one can consider archetypes of any subgroup of . In you have the further normal subgroup , that is the 10-element subgroup of . In the language of semi-direct products, that is by understanding as a subgroup . ${\displaystyle K^{*}}$ ${\displaystyle \mathrm {AGL} _{1}(5)}$ ${\displaystyle {\det }^{-1}(\{1,4\})}$ ${\displaystyle \mathrm {AGL} _{1}(5)}$ ${\displaystyle \mathbb {Z} _{5}\rtimes \mathbb {Z} _{2}}$ ${\displaystyle \mathbb {Z} _{2}}$ ${\displaystyle \{1,4\}\subset \mathbb {Z} _{5}^{*}}$

The affine semilinear group

The affine groups arise from the general, linear group by adding the translations. These groups can be further enlarged by adding body automorphisms. Is an automorphism on , and , so be ${\displaystyle \mathrm {AGL} _{n}(K)}$ ${\displaystyle \mathrm {GL} _{n}(K)}$ ${\displaystyle \sigma }$ ${\displaystyle K}$ ${\displaystyle A\in \mathrm {GL} _{n}(K)}$ ${\displaystyle v\in K^{n}}$

${\displaystyle f_{A,v,\sigma }:K^{n}\rightarrow K^{n},\,x\mapsto A(\sigma (x))+v}$,

where is defined by component-wise application to the components of the column vector . Such mappings are called affine-semilinear, compositions and inversions of bijective affine-semilinear mappings are again of this kind. ${\displaystyle \sigma (x)}$ ${\displaystyle x\in K^{n}}$

${\displaystyle \mathrm {A\Gamma L} _{n}(K):=\{f_{A,v,\sigma };\,A\in \mathrm {GL} _{n}(K),v\in K^{n},\sigma \in \mathrm {Aut} (K)\}}$

is called an affine semilinear group .

In the cases or with a prime number there are no non-trivial body automorphisms and nothing new is obtained. With bodies such as , one is dealing with real extensions of . If one understands the affine mappings as structure-preserving mappings of affine spaces , then in general the full automorphism group is not the affine structure; this is only obtained through the generally larger affine semilinear group. ${\displaystyle K=\mathbb {Q} }$ ${\displaystyle K=\mathbb {Z} _{p}}$ ${\displaystyle p}$ ${\displaystyle K=GF(4)}$ ${\displaystyle \mathrm {AGL} _{n}(K)}$ ${\displaystyle \mathrm {AGL} _{n}(K)}$

Individual evidence

1. ↑ JD Dixon, B. Mortimer: Permutation Groups , Springer-Verlag (1996), ISBN 0-387-94599-7 , chap. 2.8: Affine and Projective Groups
2. ↑ M. Schottenloher: Geometry and Symmetry in Physics , Springer-Verlag (1995), ISBN 978-3-528-06565-2 , page 27
3. ^ R. Walter: Lineare Algebra und Analytische Geometrie , Vieweg (1985), ISBN 978-3-528-08584-1 , page 168
4. ↑ B. Huppert: Finite Groups I , Springer-Verlag (1967), Chapter II, §6, Proposition 6.2
5. ^ W. Kühnel: Matrices and Lie groups , Teubner-Verlag 2011, ISBN 978-3-8348-1365-7 , Lemma 5.3
6. ↑ JD Dixon, B. Mortimer: Permutation Groups , Springer-Verlag (1996), ISBN 0-387-94599-7 , chap. 2.8: Affine and Projective Groups , page 54