example of multiply transitive
Theorem 1.
-
1.
The general linear group
acts transitively on the set of points (1-dimensional subspaces
) in the projective geometry
.
-
2.
is doubly transitive on the set of all of points in .
-
3.
is not 3-transitive on the set of all points in if .
Proof.
Evidently 2 implies 1. So suppose we have pairs of distinct points and . Then take
, , and .
As , and are linearly independent, just as and are. Therefore extending
to a basis and to a basis , we know there is a linear transformation
taking to – consider the change of basis matrix. Therefore is
2-transitive.
Now suppose . Then there exists a linearly indepedent set which
gives three distinct non-collinear points , ,
and . But then we also have three collinear points
where . As prevserves the geometry of ,
we cannot have a map in send to .
∎
Note that the action of on is not faithful so we use instead .
Title | example of multiply transitive |
---|---|
Canonical name | ExampleOfMultiplyTransitive |
Date of creation | 2013-03-22 17:21:56 |
Last modified on | 2013-03-22 17:21:56 |
Owner | Algeboy (12884) |
Last modified by | Algeboy (12884) |
Numerical id | 4 |
Author | Algeboy (12884) |
Entry type | Example |
Classification | msc 20B20 |