Examples of groups

Projective general linear group

The projective general linear group is the group of homographies on , i.e. the quotient group group

corresponding to the induced action of the general linear group of a vector space on the associated to the Projective space . Here is the Centre of the general linear group consisting of scalar matrices.

Properties

Let , and denote

1. acts on as a Collineätion, and as such, forms a subgroup of the Projective semilinear group .¹
2. acts regularly on the set of -tuples of points in general position.²

Proof of 1–2

That each induces a bijection follows from it being a bijection on . That it preserves incidence follows from the fact that it preserves linear combinations, proving ^P1.

Let and be ordered tuples of points in general position. It follows that there exist representative vectors and such that and , since each set of vectors must form a basis of . It follows there exists a linear automorphism giving the corresponding change of basis, wherefore

which proves transitivity. Suppose maps to itself. Then each of is an eigenvector of , so by the Scalar transformation criterion is a scalar transformation and thus is the identity, proving freeness. Hence the action is regular, proving ^P2.

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tidy | en | sembr

Footnotes

1. 2020. Finite geometries, ¶4.9, p. 81 ↩

2. 2020. Finite geometries, ¶4.16, p. 84 ↩