Projective linear group

Explicitly: where SL(V) is the special linear group over V and SZ(V) is the subgroup of scalar transformations with unit determinant.

Here SZ is the center of SL, and is naturally identified with the group of nth roots of unity in F (where n is the dimension of V and F is the base field).

PGL and PSL are some of the fundamental groups of study, part of the so-called classical groups, and an element of PGL is called projective linear transformation, projective transformation or homography.

If V is the n-dimensional vector space over a field F, namely V = Fn, the alternate notations PGL(n, F) and PSL(n, F) are also used.

One can define a projective space axiomatically in terms of an incidence structure (a set of points P, lines L, and an incidence relation I specifying which points lie on which lines) satisfying certain axioms – an automorphism of a projective space thus defined then being an automorphism f of the set of points and an automorphism g of the set of lines, preserving the incidence relation,[note 2] which is exactly a collineation of a space to itself.

Projective linear transforms are collineations (planes in a vector space correspond to lines in the associated projective space, and linear transforms map planes to planes, so projective linear transforms map lines to lines), but in general not all collineations are projective linear transforms – PGL is in general a proper subgroup of the collineation group.

For n ≥ 3, the collineation group is the projective semilinear group, PΓL – this is PGL, twisted by field automorphisms; formally, PΓL ≅ PGL ⋊ Gal(K / k), where k is the prime field for K; this is the fundamental theorem of projective geometry.

Thus for K a prime field (Fp or Q), we have PGL = PΓL, but for K a field with non-trivial Galois automorphisms (such as Fpn for n ≥ 2 or C), the projective linear group is a proper subgroup of the collineation group, which can be thought of as "transforms preserving a projective semi-linear structure".

However, with the exception of the non-Desarguesian planes, all projective spaces are the projectivization of a linear space over a division ring though, as noted above, there are multiple choices of linear structure, namely a torsor over Gal(K / k) (for n ≥ 3).

), while rotations in axes parallel to the hyperplane are proper projective maps, and accounts for the remaining n dimensions.

As for Möbius transformations, the group PGL(2, K) can be interpreted as fractional linear transformations with coefficients in K. Points in the projective line over K correspond to pairs from K2, with two pairs being equivalent when they are proportional.

These exceptional isomorphisms can be understood as arising from the action on the projective line.

[3] Galois constructed them as fractional linear transforms, and observed that they were simple except if p was 2 or 3; this is contained in his last letter to Chevalier.

[4] In the same letter and attached manuscripts, Galois also constructed the general linear group over a prime field, GL(ν, p), in studying the Galois group of the general equation of degree pν.

The groups PSL(n, q) (general n, general finite field) for any prime power q were then constructed in the classic 1870 text by Camille Jordan, Traité des substitutions et des équations algébriques.

The "O" is for big O notation, meaning "terms involving lower order".

The groups over F5 have a number of exceptional isomorphisms: They can also be used to give a construction of an exotic map S5 → S6, as described below.

To understand these maps, it is useful to recall these facts: Thus the image is a 3-transitive subgroup of known order, which allows it to be identified.

These three exceptional cases are also realized as the geometries of polyhedra (equivalently, tilings of Riemann surfaces), respectively: the compound of five tetrahedra inside the icosahedron (sphere, genus 0), the order 2 biplane (complementary Fano plane) inside the Klein quartic (genus 3), and the order 3 biplane (Paley biplane) inside the buckyball surface (genus 70).

More surprisingly, the coset space L2(11) / (Z / 11Z), which has order 660/11 = 60 (and on which the icosahedral group acts) naturally has the structure of a buckeyball, which is used in the construction of the buckyball surface.

The groups PSL(2, Z / nZ) arise in studying the modular group, PSL(2, Z), as quotients by reducing all elements mod n; the kernels are called the principal congruence subgroups.

The subgroup can be expressed as fractional linear transformations, or represented (non-uniquely) by matrices, as: Note that the top row is the identity and the two 3-cycles, and are orientation-preserving, forming a subgroup in PSL(2, Z), while the bottom row is the three 2-cycles, and are in PGL(2, Z) and PSL(2, Z[i]), but not in PSL(2, Z), hence realized either as matrices with determinant −1 and integer coefficients, or as matrices with determinant 1 and Gaussian integer coefficients.

That is, the subgroup C3 < S3 consisting of the identity and the 3-cycles, {(), (0 1 ∞), (0 ∞ 1)}, fixes these two points, while the other elements interchange them.

This corresponds to the action of S3 on the 2-cycles (its Sylow 2-subgroups) by conjugation and realizes the isomorphism with the group of inner automorphisms, S3 ~→ Inn(S3) ≅ S3.

Over the real and complex numbers, the topology of PGL and PSL can be determined from the fiber bundles that define them: via the long exact sequence of a fibration.

Over the real and complex numbers, the projective special linear groups are the minimal (centerless) Lie group realizations for the special linear Lie algebra

These were studied by Issai Schur, who showed that projective representations of G can be classified in terms of linear representations of central extensions of G. This led to the Schur multiplier, which is used to address this question.

The projective linear group is mostly studied for n ≥ 2, though it can be defined for low dimensions.

Thus, PGL(1, K) is the trivial group, consisting of the unique map from a singleton set to itself.