example of multiply transitive

Theorem 1.
  1. 1.

    The general linear groupMathworldPlanetmath GL(V) acts transitively on the set of points (1-dimensional subspacesPlanetmathPlanetmathPlanetmath) in the projective geometryMathworldPlanetmath 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 dimV2.


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 PQ, x and y are linearly independentMathworldPlanetmath, 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 fGL(V) taking B to C – consider the change of basis matrix. Therefore GL(V) is 2-transitive.

Now suppose dimV2. Then there exists a linearly indepedent set {x,y,z} which gives 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 geometryMathworldPlanetmath 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 faithfulPlanetmathPlanetmath 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