Projective linear group

The projective linear groups in mathematics studied groups resulting from the general linear group are constructed. If the underlying body is finite, then important finite groups are obtained ; if the body is or , one obtains Lie groups in this way . Closely related are the special projective groups that lead to an infinite series of simple groups . ${\displaystyle \mathbb {R} }$ ${\displaystyle \mathbb {C} }$

Definitions

Let it be a vector space over the body . The general linear group is the group of linear automorphisms . The center of this group is the set of the scalar multiples of the identical mapping other than 0 , that is ${\displaystyle V}$ ${\displaystyle K}$ ${\displaystyle \mathrm {GL} (V)}$ ${\displaystyle V\to V}$ ${\displaystyle \mathrm {Z} (\mathrm {GL} (V))}$ ${\displaystyle \mathrm {id} _{V}}$

${\displaystyle \mathrm {Z} (\mathrm {GL} (V))=K^{\times }\cdot \mathrm {id} _{V}}$.

Since the center is a normal divisor , one can use the factor group

${\displaystyle \mathrm {PGL} (V):=\mathrm {GL} (V)/\mathrm {Z} (\mathrm {GL} (V))}$

form. This group is called the projective linear group on . ${\displaystyle V}$

If an n -dimensional vector space is over , so , one writes ${\displaystyle V}$ ${\displaystyle K}$ ${\displaystyle V\cong K^{n}}$

${\displaystyle \mathrm {PGL} _{n}(K)}$or for . ${\displaystyle \mathrm {PGL} (n,K)}$ ${\displaystyle \mathrm {PGL} (K^{n})}$

If the finite field with , prime number , elements is clearly determined except for isomorphism , then one writes ${\displaystyle K}$ ${\displaystyle q=p^{k}}$ ${\displaystyle p}$

${\displaystyle \mathrm {PGL} _{n}(q)}$or for . ${\displaystyle \mathrm {PGL} (n,q)}$ ${\displaystyle \mathrm {PGL} _{n}(K)}$

In the case of finite-dimensional vector spaces , the determinant function is a group homomorphism ${\displaystyle V=K^{n}}$

${\displaystyle \mathrm {det} \colon \mathrm {GL} _{n}(K)\rightarrow K^{\times }}$.

The core of this homomorphism is called the special linear group . If the construction described above is restricted to this, one obtains ${\displaystyle \mathrm {SL} _{n}(K):={\det }^{-1}(\{1\})}$

${\displaystyle \mathrm {PSL} _{n}(K):=\mathrm {SL} _{n}(K)/\mathrm {Z} (\mathrm {SL} _{n}(K))}$,

the so-called projective special linear group, or for shorter the projective special or special projective group. Here is the center

${\displaystyle \mathrm {Z} (\mathrm {SL} _{n}(K))=\{\lambda \cdot \mathrm {id} _{K^{n}}\mid \lambda \in \mathrm {E} _{n}\}}$,

where is the set of -th roots of unity of . If the body is again with elements, one writes ${\displaystyle \mathrm {E} _{n}\subset K^{\times }}$ ${\displaystyle n}$ ${\displaystyle K}$ ${\displaystyle K}$ ${\displaystyle q=p^{k}}$

${\displaystyle \mathrm {PSL} _{n}(q)}$or for . ${\displaystyle \mathrm {PSL} (n,q)}$ ${\displaystyle \mathrm {PSL} _{n}(K)}$

Origin of name

Let it be the -dimensional vector space over the body . It is well known that the set of all one-dimensional subspaces is called the projective space . Each matrix from maps one-dimensional subspaces onto those again, this operation of two matrices on the same is when the matrices differ only by a scalar multiple, i.e. by one element from the center . The reverse is also true, because if two matrices permute the one-dimensional subspaces in the same way, then leaves all one-dimensional subspaces fixed, that is, every vector is an eigenvector of . Hence the only eigenspace to an eigenvalue and one has ${\displaystyle K^{n}}$ ${\displaystyle n}$ ${\displaystyle K}$ ${\displaystyle KP^{n-1}}$ ${\displaystyle \mathrm {GL} _{n}(K)}$ ${\displaystyle KP^{n-1}}$ ${\displaystyle \mathrm {Z} (\mathrm {GL} _{n}(K))}$ ${\displaystyle A,B\in \mathrm {GL} _{n}(K)}$ ${\displaystyle A^{-1}B}$ ${\displaystyle A^{-1}B}$ ${\displaystyle K^{n}}$ ${\displaystyle a}$

${\displaystyle A^{-1}B=a\cdot \mathrm {id} _{K^{n}}\in \mathrm {Z} (\mathrm {GL} _{n}(K))}$.

From this it follows that operates faithfully on projective space . This suggests the term projective linear group . ${\displaystyle \mathrm {PGL} _{n}(K)=\mathrm {GL} _{n}(K)/\mathrm {Z} (\mathrm {GL} _{n}(K))}$

Finite groups

In the following we assume a field with , prime, elements. As is well known, there is only one such body, except for isomorphism, and every finite body is of this type. The finiteness of the body results in the finiteness of , because there are only matrices with columns and rows above , and so the finiteness of the projective follows linear group and the special projective group . A closer look shows: ${\displaystyle K}$ ${\displaystyle q=p^{k}}$ ${\displaystyle p}$ ${\displaystyle \mathrm {GL} _{n}(K)}$ ${\displaystyle q^{n^{2}}}$ ${\displaystyle n}$ ${\displaystyle K}$ ${\displaystyle \mathrm {PGL} _{n}(q)}$ ${\displaystyle \mathrm {PSL} _{n}(q)}$

${\displaystyle \mathrm {PGL} _{n}(q)}$   has     elements. ${\displaystyle (q^{n}-1)\cdot (q^{n}-q)\cdot \ldots \cdot (q^{n}-q^{n-2})\cdot q^{n-1}}$

${\displaystyle \mathrm {PSL} _{n}(q)}$   has     elements. ${\displaystyle (q^{n}-1)\cdot (q^{n}-q)\cdot \ldots \cdot (q^{n}-q^{n-2})\cdot q^{n-1}/\mathrm {ggT} (n,q-1)}$

Note that one does not have to distinguish between and for the field with 2 elements , since the determinant in this case is the trivial group homomorphism. In this case, too, the center is only one element, and one has ${\displaystyle K}$ ${\displaystyle \mathrm {PGL} _{n}(2)}$ ${\displaystyle \mathrm {PSL} _{n}(2)}$ ${\displaystyle \mathrm {Z} (\mathrm {GL} _{n}(K))=K^{\times }\cdot \mathrm {id} _{K^{n}}}$

${\displaystyle \mathrm {GL} _{n}(K)\cong \mathrm {PGL} _{n}(2)\cong \mathrm {PSL} _{n}(2)}$.

simplicity

The most important property of the special projective groups is their simplicity :

• With the exception of and , the groups are simple. ${\displaystyle \mathrm {PSL} _{2}(2)}$ ${\displaystyle \mathrm {PSL} _{2}(3)}$ ${\displaystyle \mathrm {PSL} _{n}(q)}$

The special projective groups thus form one of the series of simple groups from the classification set of finite simple groups . More precisely, it is the series of simple groups of Lie type A _n , it is . ${\displaystyle \mathrm {PSL} (n,q)\cong A_{n-1}(q)}$

Isomorphisms

The following isomorphisms exist among the small special projective groups and the symmetrical groups and alternating groups : ${\displaystyle S_{n}}$ ${\displaystyle A_{n}}$

${\displaystyle \mathrm {PSL} _{2}(2)\cong \mathrm {SL} _{2}(2)\cong \mathrm {GL} _{2}(2)\cong S_{3}}$   (see page ₃ )

${\displaystyle \mathrm {PSL} _{2}(3)\cong A_{4}}$   (see A ₄ )

${\displaystyle \mathrm {PSL} _{2}(4)\cong \mathrm {PSL} _{2}(5)\cong A_{5}}$(see A ₅ )

${\displaystyle \mathrm {PSL} _{2}(7)\cong \mathrm {PSL} _{3}(2)}$

${\displaystyle \mathrm {PSL} _{4}(2)\cong \mathrm {GL} _{4}(2)\cong A_{8}}$

${\displaystyle \mathrm {PSL} _{2}(9)\cong A_{6}}$

There are no other isomorphisms between the special projective, symmetrical and alternating groups.

The smallest of these simple groups that is not alternating is therefore that , a group with 168 elements. It is actually behind the second smallest non-Abelian simple group. ${\displaystyle \mathrm {PSL} _{2}(7)}$ ${\displaystyle A_{5}}$

We also has elements, but is not isomorphic to . ${\displaystyle \mathrm {PSL} _{4}(2)}$ ${\displaystyle \mathrm {PSL} _{3}(4)}$ ${\displaystyle \textstyle 20160=8!/2}$ ${\displaystyle A_{8}}$

Broken linear transformations

In the two-dimensional case , the elements of the group can be understood as broken linear transformations. Is ${\displaystyle \mathrm {PSL} _{2}(K)}$

${\displaystyle {\begin{pmatrix}a&b\\c&d\end{pmatrix}}\in \mathrm {GL} _{2}(K)}$   with determinant   , ${\displaystyle ad-bc=1}$

so consider the fractional linear transformation

${\displaystyle x\mapsto {\frac {ax+b}{cx+d}}}$.

The set of broken linear transformations forms a group with regard to the execution one after the other as a link and the above assignment

${\displaystyle {\begin{pmatrix}a&b\\c&d\end{pmatrix}}\quad \mapsto \quad \left(x\mapsto {\frac {ax+b}{cx+d}}\right)}$

is a homomorphism of to the group of fractional linear transformations, the core of which is the center. Therefore, the group can alternatively be viewed as a group of broken linear transformations. ${\displaystyle \mathrm {SL} _{2}(K)}$ ${\displaystyle \mathrm {PSL} _{2}(K)}$

The determinant condition can be weakened to the effect that a square is, which is always the case in the body . Is , so is , because it is a question of the determinant of an invertible matrix, and . The matrix is evidently mapped onto the same fractional linear transformation. ${\displaystyle ad-bc=1}$ ${\displaystyle ad-bc}$ ${\displaystyle \mathbb {C} }$ ${\displaystyle ad-bc=r^{2}}$ ${\displaystyle r\not =0}$ ${\displaystyle (ar^{-1})(dr^{-1})-(br^{-1})(cr^{-1})=1}$ ${\displaystyle {\begin{pmatrix}ar^{-1}&br^{-1}\\cr^{-1}&dr^{-1}\end{pmatrix}}}$

Is expanded by the projective line , the elements of the one-dimensional sub-spaces and are, and is defined at the broken linear transformations, as usual, a division by 0 as and a division by a 0, the operation corresponding to the on to the operation of the fractional linear transformations . ${\displaystyle K}$ ${\displaystyle \infty }$ ${\displaystyle KP^{1}}$ ${\displaystyle K\cdot (1,a)\leftrightarrow a\in K}$ ${\displaystyle K\cdot (0,1)\leftrightarrow \infty }$ ${\displaystyle \infty }$ ${\displaystyle \infty }$ ${\displaystyle \mathrm {PSL} _{2}(K)}$ ${\displaystyle KP^{1}}$ ${\displaystyle KP^{1}}$

Lie groups

Is or so you get Lie groups or and the special groups or . The latter are of type A _n-1 for the Lie groups for Lie algebra . The description as a broken linear transformation is also called Möbius transformation . ${\displaystyle K=\mathbb {R} }$ ${\displaystyle K=\mathbb {C} }$ ${\displaystyle \mathrm {PGL} _{n}(\mathbb {R} )}$ ${\displaystyle \mathrm {PGL} _{n}(\mathbb {C} )}$ ${\displaystyle \mathrm {PSL} _{n}(\mathbb {R} )}$ ${\displaystyle \mathrm {PSL} _{n}(\mathbb {C} )}$ ${\displaystyle n\geq 2}$

Individual evidence

1. ↑ B. Huppert: Finite Groups I. Springer-Verlag (1967), Chapter II, Proposition 6.2.
2. ^ B. Huppert: Endliche Gruppen I. Springer-Verlag (1967), Chapter II, main clause 6.13.
3. ^ Roger W. Carter: Simple Groups of Lie Type. John Wiley & Sons 1972, ISBN 0-471-13735-9 , Theorem 11.3.2. (I).
4. ↑ B. Huppert: Endliche Gruppen I. Springer-Verlag (1967), Chapter II, Sentence 6.14.
5. ^ DJS Robinson : A Course in the Theory of Groups. Springer-Verlag 1996, ISBN 978-1-4612-6443-9 , page 78.
6. ^ B. Huppert: Endliche Gruppen I. Springer-Verlag (1967), Chapter II, Proposition 8.1.
7. ^ P. Anglès: Conformal Groups in Geometry and Spin Structures. Springer-Verlag 2007, ISBN 978-0-8176-3512-1 , chap. 1.1: Classical Groups.