Projective general linear group

The projective general linear group $P G L (𝑉)$ is the group of homographies on $P (𝑉)$ , i.e. the quotient group #m/def/group

P G L (𝑉) = G L (𝑉) / Z (𝑉)

corresponding to the induced action of the general linear group $G L (𝑉)$ of a vector space $𝑉$ on the associated to the Projective space $P (𝑉)$ . Here $Z (𝑉) ≅ 𝕂 \times$ is the Centre of the general linear group consisting of scalar matrices.

Properties

Let $𝑉 = 𝕂 𝑛 + 1$ , and denote $P G L (𝑛 + 1, 𝕂) = P G L (𝑉)$

$P G L (𝑛 + 1, 𝕂)$ acts on $P G (𝑛, 𝕂)$ as a Collineätion, and as such, $P G L (𝑛 + 1, 𝕂)$ forms a subgroup of the Projective semilinear group $P Γ L (𝑛, 𝕂)$ .¹
$P G L (𝑛 + 1, 𝕂)$ acts regularly on the set of $(𝑛 + 2)$ -tuples of points in general position.²

Proof of 1–2

That each $𝐴 \in G L (𝑛 + 1, 𝕂)$ induces a bijection follows from it being a bijection on $𝑉$ . That it preserves incidence follows from the fact that it preserves linear combinations, proving ^P1.

Let $(𝐴 0, \dots, 𝐴 𝑛 + 1)$ and $(𝐵 0, \dots, 𝐵 𝑛 + 1)$ be ordered tuples of points in general position. It follows that there exist representative vectors $𝐚 0, \dots, 𝐚 𝑛$ and $𝐛 0, \dots, 𝐛 𝑛$ such that $𝐴 𝑛 + 1 = [𝐚 0 + \dots + 𝐚 𝑛]$ and $𝐵 𝑛 + 1 = [𝐛 0 + \dots + 𝐛 𝑛]$ , since each set of $𝑛$ vectors must form a basis of $𝑉$ . It follows there exists a linear automorphism $Φ$ giving the corresponding change of basis, wherefore

Φ (𝐚 𝑛 + 1) = Φ (𝐚 0 + \dots + 𝐚 𝑛) = 𝐛 0 + \dots + 𝐛 𝑛 = 𝐛 𝑛 + 1

which proves transitivity. Suppose $[𝐴] \in P G L (𝑛 + 1, 𝕂)$ maps $(𝐴 0, \dots, 𝐴 𝑛 + 1)$ to itself. Then each of $𝐚 0, \dots, 𝐚 𝑛 + 1$ is an eigenvector of $𝐴$ , so by the Scalar transformation criterion $𝐴$ is a scalar transformation and thus $[𝐴]$ is the identity, proving freeness. Hence the action is regular, proving ^P2.

#state/tidy | #lang/en | #SemBr

2020. Finite geometries, ¶4.9, p. 81 ↩
2020. Finite geometries, ¶4.16, p. 84 ↩

Projective general linear group

Properties

Footnotes