example of multiply transitive

Theorem 1.

1. The general linear group $GL(V)$ acts transitively on the set of points (1-dimensional subspaces) in theprojective 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 $dimV\ne 2$.

Proof.

Evidently 2 implies 1. So suppose we have pairs of distinct points $(P,Q)$ and $(R,S)$. Then take$P=⟨x⟩$, $Q=⟨y⟩$, $R=⟨z⟩$ and $S=⟨w⟩$.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 $GL(V)$ is2-transitive.

Now suppose $dimV\ge 2$. Then there exists a linearly indepedent set $\{x,y,z\}$ whichgives three distinct non-collinear points $(P,Q,R)$, $P=⟨x⟩$, $Q=⟨y⟩$and $R=⟨z⟩$. But then we also have three collinear points$(P,Q,S)$ where $S=⟨x+y⟩$. 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 $PGL(V)$.