example of multiply transitive

Theorem 1.
1. 1.

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

2. 2.

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

3. 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 ExampleOfMultiplyTransitive 2013-03-22 17:21:56 2013-03-22 17:21:56 Algeboy (12884) Algeboy (12884) 4 Algeboy (12884) Example msc 20B20