Prove PGL(V) Acts 2-Transitively on P(V) Projective Space

Thread starter gankutsuou7
Start date Nov 28, 2010
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Algebra Geometric Geometric algebra

In summary, PGL(V) acting 2-transitively on P(V) projective space means that it can freely move any point in P(V) to any other point without changing its projective structure. This is significant because it characterizes PGL(V) as the automorphism group of P(V) and makes it a fundamental group in the study of projective geometry. PGL(V) is related to linear transformations as it is the quotient group of GL(V) by its center. The proof for PGL(V) acting 2-transitively involves expressing points in homogeneous coordinates and using the fact that projective transformations preserve the cross-ratio. This property is closely related to the group's structure and plays a crucial role in

Nov 28, 2010

gankutsuou7

Dim(V)>1.Prove that PGL(V) acts two transitively but not 3 transitively on P(V) projective space.

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Nov 29, 2010

arkajad

Idea: Any two non-colinear vectors form a basis. Three vectors form a basis only when they are linearrly independent. You can't map linearly independent vectors to linearly dependent vectors by a GL(V) transformation.

Related to Prove PGL(V) Acts 2-Transitively on P(V) Projective Space

1. What does it mean for PGL(V) to act 2-transitively on P(V) projective space?

Acting 2-transitively means that for any two distinct points in projective space P(V), there exists a unique projective transformation in the group PGL(V) that can map one point to the other. In other words, PGL(V) can freely move any point in P(V) to any other point without changing its projective structure.

2. What is the significance of PGL(V) acting 2-transitively on P(V) projective space?

This property is significant because it characterizes the group PGL(V) as the automorphism group of projective space P(V). This means that any projective transformation on P(V) can be achieved by composing elements of PGL(V), making it a fundamental group in the study of projective geometry.

3. How is PGL(V) related to linear transformations?

PGL(V) is the quotient group of the general linear group GL(V) by its center, which consists of scalar multiples of the identity matrix. This means that PGL(V) contains all the linear transformations of the vector space V, but with the added restriction that scalar multiples are considered to be the same transformation. In other words, PGL(V) captures the essential geometric properties of linear transformations without considering their magnitudes.

4. What is the proof that PGL(V) acts 2-transitively on P(V) projective space?

The proof involves showing that for any two distinct points in P(V), there exists a unique projective transformation in PGL(V) that maps one point to the other. This is done by expressing the points in homogeneous coordinates and using the fact that projective transformations preserve the cross-ratio of four collinear points. By constructing a projective transformation that maps the first point to the second while preserving the cross-ratio, it can be shown that this transformation is unique and belongs to PGL(V).

5. How is the 2-transitive action of PGL(V) on P(V) related to the group's properties?

The 2-transitive action of PGL(V) on P(V) is one of the defining properties of this group. It is closely related to the group's structure and other important properties, such as its normal subgroups and its isomorphisms with other groups. Additionally, this property plays a crucial role in the study of projective geometry and the classification of projective spaces.