example of multiply transitive

Theorem 1.

1.

The general linear group $GL(V)$ acts transitively on the set of points (1-dimensional subspaces) in the projective geometry $PG(V)$ .
2.

$PGL(V)$ is doubly transitive on the set of all of points in $PG(V)$ .
3.

$PGL(V)$ is not 3-transitive on the set of all points in $PG(V)$ if $\dim V\neq 2$ .

Proof.

Evidently 2 implies 1. So suppose we have pairs of distinct points $(P,Q)$ and $(R,S)$ . Then take $P=\langle x\rangle$ , $Q=\langle y\rangle$ , $R=\langle z\rangle$ and $S=\langle w\rangle$ . As $P\neq Q$ , $x$ and $y$ are linearly independent, just as $z$ and $w$ are. Therefore extending $\{x,y\}$ to a basis $B$ and $\{z,w\}$ to a basis $C$ , we know there is a linear transformation $f\in GL(V)$ taking $B$ to $C$ – consider the change of basis matrix. Therefore $GL(V)$ is 2-transitive.

Now suppose $\dim V\geq 2$ . Then there exists a linearly indepedent set $\{x,y,z\}$ which gives three distinct non-collinear points $(P,Q,R)$ , $P=\langle x\rangle$ , $Q=\langle y\rangle$ and $R=\langle z\rangle$ . But then we also have three collinear points $(P,Q,S)$ where $S=\langle x+y\rangle$ . As $GL(V)$ prevserves the geometry of $PG(V)$ , we cannot have a map in $GL(V)$ send $(P,Q,R)$ to $(P,Q,S)$ . ∎

Note that the action of $GL(V)$ on $PG(V)$ is not faithful so we use instead $PGL(V)$ .

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