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 .
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
-
.
Since the center is a normal divisor , one can use the factor group
form. This group is called the projective linear group on .
If an n -dimensional vector space is over , so , one writes
-
or for .
If the finite field with , prime number , elements is clearly determined except for isomorphism , then one writes
-
or for .
In the case of finite-dimensional vector spaces , the determinant function is a group homomorphism
-
.
The core of this homomorphism is called the special linear group . If the construction described above is restricted to this, one obtains
-
,
the so-called projective special linear group, or for shorter the projective special or special projective group. Here is the center
-
,
where is the set of -th roots of unity of . If the body is again with elements, one writes
-
or for .
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
-
.
From this it follows that operates faithfully on projective space . This suggests the term projective linear group .
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:
-
has elements.
-
has elements.
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
-
.
simplicity
The most important property of the special projective groups is their simplicity :
- With the exception of and , the groups are simple.
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 .
Isomorphisms
The following isomorphisms exist among the small special projective groups and the symmetrical groups and alternating groups :
-
(see page 3 )
-
(see A 4 )
-
(see A 5 )
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.
We also has elements, but is not isomorphic to .
Broken linear transformations
In the two-dimensional case , the elements of the group can be understood as broken linear transformations. Is
-
with determinant ,
so consider the fractional linear transformation
-
.
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
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.
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.
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 .
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 .
Individual evidence
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↑ B. Huppert: Finite Groups I. Springer-Verlag (1967), Chapter II, Proposition 6.2.
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^ B. Huppert: Endliche Gruppen I. Springer-Verlag (1967), Chapter II, main clause 6.13.
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^ Roger W. Carter: Simple Groups of Lie Type. John Wiley & Sons 1972, ISBN 0-471-13735-9 , Theorem 11.3.2. (I).
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↑ B. Huppert: Endliche Gruppen I. Springer-Verlag (1967), Chapter II, Sentence 6.14.
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^ DJS Robinson : A Course in the Theory of Groups. Springer-Verlag 1996, ISBN 978-1-4612-6443-9 , page 78.
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^ B. Huppert: Endliche Gruppen I. Springer-Verlag (1967), Chapter II, Proposition 8.1.
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^ P. Anglès: Conformal Groups in Geometry and Spin Structures. Springer-Verlag 2007, ISBN 978-0-8176-3512-1 , chap. 1.1: Classical Groups.