Finite collineation groups


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In all but the last case, PSL 2, 11 , it corresponds to an exceptional isomorphism, where the right-most group has an obvious action on p points:. Further, L 2 7 and L 2 11 have two inequivalent actions on p points; geometrically this is realized by the action on a biplane, which has p points and p blocks — the action on the points and the action on the blocks are both actions on p points, but not conjugate they have different point stabilizers ; they are instead related by an outer automorphism of the group. 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 The same is true for subgroups of L 2 7 isomorphic to S 4 , and this also has a biplane geometry.

Geometrically, this action can be understood via a biplane geometry , which is defined as follows. A biplane geometry is a symmetric design a set of points and an equal number of "lines", or rather blocks such that any set of two points is contained in two lines, while any two lines intersect in two points; this is similar to a finite projective plane, except that rather than two points determining one line and two lines determining one point , they determine two lines respectively, points.

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In this case the Paley biplane , obtained from the Paley digraph of order 11 , the points are the affine line the finite field F 11 , where the first line is defined to be the five non-zero quadratic residues points which are squares: 1, 3, 4, 5, 9 , and the other lines are the affine translates of this add a constant to all the points. L 2 11 is then isomorphic to the subgroup of S 11 that preserve this geometry sends lines to lines , giving a set of 11 points on which it acts — in fact two: the points or the lines, which corresponds to the outer automorphism — while L 2 5 is the stabilizer of a given line, or dually of a given point.

The group PSL 3, 4 can be used to construct the Mathieu group M 24 , one of the sporadic simple groups ; in this context, one refers to PSL 3, 4 as M 21 , though it is not properly a Mathieu group itself. One begins with the projective plane over the field with four elements, which is a Steiner system of type S 2, 5, 21 — meaning that it has 21 points, each line "block", in Steiner terminology has 5 points, and any 2 points determine a line — and on which PSL 3, 4 acts.

PSL groups arise as Hurwitz groups automorphism groups of Hurwitz surfaces — algebraic curves of maximal possibly symmetry group.


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The Hurwitz surface of lowest genus, the Klein quartic genus 3 , has automorphism group isomorphic to PSL 2, 7 equivalently GL 3, 2 , while the Hurwitz surface of second-lowest genus, the Macbeath surface genus 7 , has automorphism group isomorphic to PSL 2, 8. In fact, many but not all simple groups arise as Hurwitz groups including the monster group , though not all alternating groups or sporadic groups , though PSL is notable for including the smallest such groups.

The subgroup can be expressed as fractional linear transformations , or represented non-uniquely by matrices, as:. Over the real and complex numbers, the topology of PGL and PSL can be determined from the fiber bundles that define them:. For both the reals and complexes, SL is a covering space of PSL, with number of sheets equal to the number of n th roots in K ; thus in particular all their higher homotopy groups agree.

Finite collineation groups

Conversely, its universal covering group is the maximal simply connected element, and the intermediary realizations form a lattice of covering groups. 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. Thus, PGL 0, K is the trivial group, consisting of the unique empty map from the empty set to itself.

Thus, PGL 1, K is the trivial group, consisting of the unique map from a singleton set to itself. From Wikipedia, the free encyclopedia.

For other uses, see Projective group disambiguation. Classical groups. Simple Lie groups. The method relies on DeltaOminus , and computes the generators to be added. Furthermore, the generators to be added are computed. The method relies on DeltaOplus , and computes the generators to be added. The generators of this group are described exaplicitely in [VM98] , Appendix D.

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Projective linear group

A correction can be found in [Off00]. However, also this source contains a mistake. From a geometrical point of view, it is straightforward to construct a basis for a collineation group for the action on projective points. The returned record contains the base points for the action, and some other fields.

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The information in the other fields is determined from the arguments opt and i. More information on these details can be found in the manual of GenSS. Click on an option below to access. Log out of ReadCube.

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