Projective Linear Group - Properties

Properties

• PGL sends collinear points to collinear points (it preserves projective lines), but it is not the full collineation group, which is instead either PΓL (for ) or the full symmetric group for (the projective line).
• Every (biregular) algebraic automorphism of a projective space is projective linear. The birational automorphisms form a larger group, the Cremona group.
• PGL acts faithfully on projective space: non-identity elements act non-trivially.

  Concretely, the kernel of the action of GL on projective space is exactly the scalar maps, which are quotiented out in PGL.

• PGL acts 2-transitively on projective space.

  This is because 2 distinct points in projective space correspond to 2 vectors that do not lie on a single linear space, and hence are linearly independent, and GL acts transitively on k-element sets of linearly independent vectors.

• PGL(2, K) acts sharply 3-transitively on the projective line.

  3 arbitrary points are conventionally mapped to in alternative notation, In fractional linear transformation notation, the function maps and is the unique such map that does so. This is the cross-ratio – see cross-ratio: transformational approach for details.

• For PGL(n, K) does not act 3-transitively, because it must send 3 collinear points to 3 other collinear points, not an arbitrary set. For the space is the projective line, so all points are collinear and this is no restriction.
• PGL(2, K) does not act 4-transitively on the projective line (except for as has 3+1=4 points, so 3-transitive implies 4-transitive); the invariant that is preserved is the cross ratio, and this determines where every other point is sent: specifying where 3 points are mapped determines the map. Thus in particular it is not the full collineation group of the projective line (except for and ).
• PSL(2, q) and PGL(2, q) (for q>2, and q odd for PSL) are two of the four families of Zassenhaus groups.
• PSL(n, K) and PGL(n, K) are algebraic groups of dimension – they are both open subgroups of the projective space

  For PGL, the is the dimension of GL(n, K), and the is from projectivization.

  For PSL, is the dimension of SL, which is a covering space of PSL, so they have the same dimension. More casually, PSL differs from SL and from PGL by a finite group in each case, so the dimensions agree.

  This is also reflected in the order of the groups over finite fields, as the degree of the order as a polynomial in q: the order of PGL(n, q) is plus lower order terms.

• PSL and PGL are centerless – this is because the diagonal matrices are not only the center, but also the hypercenter (the quotient of a group by its center is not necessarily centerless).