example of multiply transitive
Theorem 1.

1.
The general linear group^{} $GL(V)$ acts transitively on the set of points (1dimensional 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 3transitive on the set of all points in $PG(V)$ if $dimV\ne 2$.
Proof.
Evidently 2 implies 1. So suppose we have pairs of distinct points $(P,Q)$ and $(R,S)$. Then take $P=\u27e8x\u27e9$, $Q=\u27e8y\u27e9$, $R=\u27e8z\u27e9$ and $S=\u27e8w\u27e9$. As $P\ne Q$, $x$ 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 2transitive.
Now suppose $dimV\ge 2$. Then there exists a linearly indepedent set $\{x,y,z\}$ which gives three distinct noncollinear points $(P,Q,R)$, $P=\u27e8x\u27e9$, $Q=\u27e8y\u27e9$ and $R=\u27e8z\u27e9$. But then we also have three collinear points $(P,Q,S)$ where $S=\u27e8x+y\u27e9$. 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  20130322 17:21:56 
Last modified on  20130322 17:21:56 
Owner  Algeboy (12884) 
Last modified by  Algeboy (12884) 
Numerical id  4 
Author  Algeboy (12884) 
Entry type  Example 
Classification  msc 20B20 