# The geometry of elation groups of a finite projective

In mathematicsespecially in the group theoretic area of algebrathe projective linear group also known as the projective general linear group or PGL is the induced action of the general linear group of a vector space V on the associated projective space P V.

Explicitly, the projective linear group is the quotient group. The projective special linear groupPSL, is defined analogously, as the induced action of the special linear group on the associated projective space.

Here SZ is the center of SL, and is naturally identified with the group of n th 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 groupsand an element of PGL is called projective linear transformationprojective transformation or homography. The name comes from projective geometrywhere the projective group acting on homogeneous coordinates x 0 : x 1 Note that unlike the general linear group, which is generally defined axiomatically as "invertible functions preserving the linear vector space structure", the projective linear group is defined constructively, as a quotient of the general linear group of the associated vector space, rather than axiomatically as "invertible functions preserving the projective linear structure".

A related group is the collineation groupwhich is defined axiomatically. A collineation is an invertible or more generally one-to-one map which sends collinear points to collinear points. 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 linesbut in general not all collineations are projective linear transforms — PGL is in general a proper subgroup of the collineation group.

One may also define collineation groups for axiomatically defined projective spaces, where there is no natural notion of a projective linear transform. The elements of the projective linear group can be understood as "tilting the plane" along one of the axes, and then projecting to the original plane, and also have dimension n.

Points in the projective line over K correspond to pairs from K 2with two pairs being equivalent when they are proportional. When the second coordinate is non-zero, a point can be represented by [ z1]. In this way successive transformations can be written as right multiplication by such matrices, and matrix multiplication can be used for the group product in PGL 2, K.

### Projective geometry

They are finite simple groups whenever n is at least 2, with two exceptions: [2] L 2 2which is isomorphic to S 3the symmetric group on 3 letters, and is solvable ; and L 2 3which is isomorphic to A 4the alternating group on 4 letters, and is also solvable. These exceptional isomorphisms can be understood as arising from the action on the projective line. The "O" is for big O notationmeaning "terms involving lower order".

Note however that GL 2, 5 is not a double cover of S 5but is rather a 4-fold cover. Thus the image is a 3-transitive subgroup of known order, which allows it to be identified.

This yields the following maps:. This can be analyzed as follows; note that for 2 and 3 the action is not faithful it is a non-trivial quotient, and the PSL group is not simplewhile for 5, 7, and 11 the action is faithful as the group is simple and the action is non-trivialand yields an embedding into S p.

In all but the last case, PSL 2, 11it 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 surfacesrespectively: the compound of five tetrahedra inside the icosahedron sphere, genus 0the order 2 biplane complementary Fano plane inside the Klein quartic genus 3and the order 3 biplane Paley biplane inside the buckyball surface genus The same is true for subgroups of L 2 7 isomorphic to S 4and this also has a biplane geometry.

Geometrically, this action can be understood via a biplane geometrywhich 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 pointthey determine two lines respectively, points.

In this case the Paley biplaneobtained from the Paley digraph of order 11the points are the affine line the finite field F 11where the first line is defined to be the five non-zero quadratic residues points which are squares: 1, 3, 4, 5, 9and 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 linesgiving 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 24one of the sporadic simple groups ; in this context, one refers to PSL 3, 4 as M 21though 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. The Hurwitz surface of lowest genus, the Klein quartic genus 3has automorphism group isomorphic to PSL 2, 7 equivalently GL 3, 2while the Hurwitz surface of second-lowest genus, the Macbeath surface genus 7has automorphism group isomorphic to PSL 2, 8.

In fact, many but not all simple groups arise as Hurwitz groups including the monster groupthough not all alternating groups or sporadic groupsthough PSL is notable for including the smallest such groups. The subgroup can be expressed as fractional linear transformationsor 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.To browse Academia. Skip to main content.

Log In Sign Up. Nicola Durante. Alessandro Siciliano. The geometry of elation groups of a finite projective space arXiv CO] 28 Feb N. Durante and A. Siciliano February 29, Abstract We study the geometry of point-orbits of elation groups with a given center and axis of a finite projective space. We show that there exists a correspondence from conjugacy classes of such groups and orbits on projective subspaces of a suitable dimension of Singer groups of projective spaces. Together with a recent result of Drudge [7] we establish the number of these elation groups.

For such collineations there is always a center z, i. In particular, if g is not the identity, then g has no fixed point besides z and the points of A. Therefore, a non-identical perspectivity has a unique axis and a unique center. The non-identical elations with axis A are just the perspectivities without any fixed point outside A.

Therefore they form, togheter with the identity, a normal subgroup of the full collineation group of the projective space fixing A. This elation group acts fix-point-free that is, semiregularly outside A. If g is any perspectivity, hgi induces a semiregular permutation group on the non-fixed points of any line not contained in the axis through the center. The motivation for us to study this problem is that this type of elation groups play an important role in finite geometries as they appear as automorphism groups of relevant objects such as unitals, blocking sets and maximal arcs in projective spaces.

We say that two additive subgroups of GF ph are equivalent if they differ by a non-zero scalar in GF ph. Lemma 2. The solutions of 1 form the underlying vector subspace H of H. Thus we have proved the following result. Proposition 2. Here we review some results on Singer groups of vector and projective spaces.

The group G s called a Singer cyclic group of V ; see, e. In the literature, the actions of Singer groups are usually considered on points and lines.

In [10], the geometric structure of Singer line orbits in PG 3, q were investigated. This was extended to Singer plane orbits also in higher dimensional projective spaces; see [9]. For the convenience of the reader we repeat the relevant material from [7] without proofs, thus making our exposition self-contained.Finite geometry is just that: geometry with finitely many points, finitely many lines, and so on.

One of the reasons why finite geometry is interesting is that some of the properties of Euclidean geometry still hold in the finite world. For example, Pascal's Theorem states that the three points of intersection of opposite sides of a hexagram inscribed in a conic lie on a line see the picture below. The same result holds in the corresponding finite plane. To be precise, we should take a finite projective plane satisfying the theorem of Desargues. Finite geometry is intimately linked with experimental design, information security, particle physics and coding theory.

Apart from being an exciting and interesting areas in combinatorics, finite geometry has many applications to algebraic geometry, group theory, codes, graphs and designs.

A building is a combinatorial and geometric structure which generalises many of the widely known geometries, such as projective spaces, polar spaces and trees. This field, initiated by Jacques Tits in the 70s as a way to understand the exceptional groups of Lie type, has many applications mainly to group theory.

Some highlights of the work of the CMSC are the following:. Bamberg, Li, and Swartz have characterised generalised quadrangles that admit locally 2-transitive symmetries. Bamberg, Giudici, and Royle have shown that every flock generalised quadrangle with an even number of points has a hemisystem. Royle and Penttila classified all of the ovoids of finite projective 3-space for all orders up to Devillers, Muhlherr, and Van Maldeghem showed that a 3-spherical building in which each rank 2 residue is connected far away from a chamber, and each rank 3 residue is simply 2-connected far away from a chamber, admits a twinning i.

The Centre uses its breadth of knowledge of permutation groups, graph theory, computation and algebraic combinatorics to study finite geometries.

## Finite geometry

Some highlights of the work of the CMSC are the following: Bamberg, Li, and Swartz have characterised generalised quadrangles that admit locally 2-transitive symmetries.

Gordon Royle. John Bamberg. Alice Devillers.Projective geometry is a topic in mathematics. It is the study of geometric properties that are invariant with respect to projective transformations.

This means that, compared to elementary geometry, projective geometry has a different setting, projective spaceand a selective set of basic geometric concepts. The basic intuitions are that projective space has more points than Euclidean spacefor a given dimension, and that geometric transformations are permitted that transform the extra points called " points at infinity " to Euclidean points, and vice versa. Properties meaningful for projective geometry are respected by this new idea of transformation, which is more radical in its effects than can be expressed by a transformation matrix and translations the affine transformations.

The first issue for geometers is what kind of geometry is adequate for a novel situation. It is not possible to refer to angles in projective geometry as it is in Euclidean geometrybecause angle is an example of a concept not invariant with respect to projective transformations, as is seen in perspective drawing.

One source for projective geometry was indeed the theory of perspective. Another difference from elementary geometry is the way in which parallel lines can be said to meet in a point at infinityonce the concept is translated into projective geometry's terms. Again this notion has an intuitive basis, such as railway tracks meeting at the horizon in a perspective drawing. See projective plane for the basics of projective geometry in two dimensions. While the ideas were available earlier, projective geometry was mainly a development of the 19th century.

This included the theory of complex projective spacethe coordinates used homogeneous coordinates being complex numbers. Several major types of more abstract mathematics including invariant theorythe Italian school of algebraic geometryand Felix Klein 's Erlangen programme resulting in the study of the classical groups were based on projective geometry. It was also a subject with many practitioners for its own sake, as synthetic geometry. Another topic that developed from axiomatic studies of projective geometry is finite geometry.

The topic of projective geometry is itself now divided into many research subtopics, two examples of which are projective algebraic geometry the study of projective varieties and projective differential geometry the study of differential invariants of the projective transformations.

Projective geometry is an elementary non- metrical form of geometry, meaning that it is not based on a concept of distance. In two dimensions it begins with the study of configurations of points and lines. That there is indeed some geometric interest in this sparse setting was first established by Desargues and others in their exploration of the principles of perspective art.

The simplest illustration of duality is in the projective plane, where the statements "two distinct points determine a unique line" i. Projective geometry can also be seen as a geometry of constructions with a straight-edge alone. For example, the different conic sections are all equivalent in complex projective geometry, and some theorems about circles can be considered as special cases of these general theorems.

During the early 19th century the work of Jean-Victor PonceletLazare Carnot and others established projective geometry as an independent field of mathematics. After much work on the very large number of theorems in the subject, therefore, the basics of projective geometry became understood.

The incidence structure and the cross-ratio are fundamental invariants under projective transformations. Projective geometry can be modeled by the affine plane or affine space plus a line hyperplane "at infinity" and then treating that line or hyperplane as "ordinary".

In a foundational sense, projective geometry and ordered geometry are elementary since they involve a minimum of axioms and either can be used as the foundation for affine and Euclidean geometry. The first geometrical properties of a projective nature were discovered during the 3rd century by Pappus of Alexandria. He made Euclidean geometrywhere parallel lines are truly parallel, into a special case of an all-encompassing geometric system.

### Fano plane

Desargues's study on conic sections drew the attention of year-old Blaise Pascal and helped him formulate Pascal's theorem. The works of Gaspard Monge at the end of 18th and beginning of 19th century were important for the subsequent development of projective geometry.

The work of Desargues was ignored until Michel Chasles chanced upon a handwritten copy during Meanwhile, Jean-Victor Poncelet had published the foundational treatise on projective geometry during In finite geometrythe Fano plane after Gino Fano is the finite projective plane of order 2.

It is the finite projective plane with the smallest possible number of points and lines: 7 points and 7 lines, with 3 points on every line and 3 lines through every point. The standard notation for this plane, as a member of a family of projective spacesis PG 2, 2 where PG stands for " projective geometry ", the first parameter is the geometric dimension and the second parameter is the order.

The Fano plane is an example of a finite incidence structureso many of its properties can be established using combinatorial techniques and other tools used in the study of incidence geometries. Since it is a projective space, algebraic techniques can also be effective tools in its study. The Fano plane can be constructed via linear algebra as the projective plane over the finite field with two elements.

One can similarly construct projective planes over any other finite field, with the Fano plane being the smallest. Using the standard construction of projective spaces via homogeneous coordinatesthe seven points of the Fano plane may be labeled with the seven non-zero ordered triples of binary digits, and This can be done in such a way that for every two points p and qthe third point on line pq has the label formed by adding the labels of p and q modulo 2.

In other words, the points of the Fano plane correspond to the non-zero points of the finite vector space of dimension 3 over the finite field of order 2. Due to this construction, the Fano plane is considered to be a Desarguesian planeeven though the plane is too small to contain a non-degenerate Desargues configuration which requires 10 points and 10 lines.

The lines of the Fano plane may also be given homogeneous coordinates, again using non-zero triples of binary digits. With this system of coordinates, a point is incident to a line if the coordinate for the point and the coordinate for the line have an even number of positions at which they both have nonzero bits: for instance, the point belongs to the linebecause they have nonzero bits at two common positions.

In terms of the underlying linear algebra, a point belongs to a line if the inner product of the vectors representing the point and line is zero. The automorphism group GL 3,2 of the group Z 2 3 is that of the Fano plane, and has order As with any incidence structure, the Levi graph of the Fano plane is a bipartite graphthe vertices of one part representing the points and the other representing the lines, with two vertices joined if the corresponding point and line are incident.

This particular graph is a connected cubic graph regular of degree 3has girth 6 and each part contains 7 vertices. It is the Heawood graphthe unique 6-cage. A collineationautomorphismor symmetry of the Fano plane is a permutation of the 7 points that preserves collinearity: that is, it carries collinear points on the same line to collinear points. Since the field has only one nonzero element, this group is isomorphic to the projective special linear group PSL 3,2 and the general linear group GL 3,2.

It is also isomorphic to PSL 2,7. As a permutation group acting on the 7 points of the plane, the collineation group is doubly transitive meaning that any ordered pair of points can be mapped by at least one collineation to any other ordered pair of points.

Collineations may also be viewed as the color-preserving automorphisms of the Heawood graph see figure. A bijection between the point set and the line set that preserves incidence is called a duality and a duality of order two is called a polarity.See what's new with book lending at the Internet Archive.

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The price for each copy is 50 cents, which includes postage. Paire Definition of a Finite Projective Plane. Preliminary Theorems. Types of Collineations in PG 2,2". Invariant Real Configurations and Their Groups. The definition and general properties of finite projective spaces together with references to the literature of the subject may be found in a paper by veblen and BUSSEY in the Transactions of the American Mathematical Society, Vol.

We give a brief summary of the analytic and synthetic definitions of a finite projective plane. Vol, IX. The points of a line are those points whose co- ordinates. Synthetically a finite projective plane may be defined as a set of elements which for suggestiveness are called points, arranged in subsets called lines and subject to the following conditions: I. The set contains a finite number, greater than one, of lines, and each line contains p"- -l points p and n integers and p a prime.

If A and B are distinct points there is one and only one line that contains A and B. All tlie points considered arc in the same plane.A finite geometry is any geometric system that has only a finite number of points.

The familiar Euclidean geometry is not finite, because a Euclidean line contains infinitely many points. A geometry based on the graphics displayed on a computer screen, where the pixels are considered to be the points, would be a finite geometry. While there are many systems that could be called finite geometries, attention is mostly paid to the finite projective and affine spaces because of their regularity and simplicity.

Projective geometry and perspective - WildTrig: Intro to Rational Trigonometry - N J Wildberger

Finite geometries may be constructed via linear algebrastarting from vector spaces over a finite field ; the affine and projective planes so constructed are called Galois geometries. Finite geometries can also be defined purely axiomatically. Most common finite geometries are Galois geometries, since any finite projective space of dimension three or greater is isomorphic to a projective space over a finite field that is, the projectivization of a vector space over a finite field.

However, dimension two has affine and projective planes that are not isomorphic to Galois geometries, namely the non-Desarguesian planes.

Similar results hold for other kinds of finite geometries. The following remarks apply only to finite planes. There are two main kinds of finite plane geometry: affine and projective. In an affine planethe normal sense of parallel lines applies. In a projective planeby contrast, any two lines intersect at a unique point, so parallel lines do not exist. Both finite affine plane geometry and finite projective plane geometry may be described by fairly simple axioms. An affine plane geometry is a nonempty set X whose elements are called "points"along with a nonempty collection L of subsets of X whose elements are called "lines"such that:.

The last axiom ensures that the geometry is not trivial either empty or too simple to be of interest, such as a single line with an arbitrary number of points on itwhile the first two specify the nature of the geometry. The simplest affine plane contains only four points; it is called the affine plane of order 2. The order of an affine plane is the number of points on any line, see below. Since no three are collinear, any pair of points determines a unique line, and so this plane contains six lines.

It corresponds to a tetrahedron where non-intersecting edges are considered "parallel", or a square where not only opposite sides, but also diagonals are considered "parallel".

The affine plane of order 3 is known as the Hesse configuration. A projective plane geometry is a nonempty set X whose elements are called "points"along with a nonempty collection L of subsets of X whose elements are called "lines"such that:. An examination of the first two axioms shows that they are nearly identical, except that the roles of points and lines have been interchanged.

This suggests the principle of duality for projective plane geometries, meaning that any true statement valid in all these geometries remains true if we exchange points for lines and lines for points. The smallest geometry satisfying all three axioms contains seven points.

In this simplest of the projective planes, there are also seven lines; each point is on three lines, and each line contains three points. This particular projective plane is sometimes called the Fano plane. If any of the lines is removed from the plane, along with the points on that line, the resulting geometry is the affine plane of order 2.

The Fano plane is called the projective plane of order 2 because it is unique up to isomorphism. A permutation of the Fano plane's seven points that carries collinear points points on the same line to collinear points is called a collineation of the plane. One major open question in finite geometry is:. Planes not derived from finite fields also exist e. The best general result to date is the Bruck—Ryser theorem ofwhich states:. The non-existence of a finite plane of order 10 was proven in a computer-assisted proof that finished in — see Lam for details.

The next smallest number to consider is 12, for which neither a positive nor a negative result has been proved. Individual examples can be found in the work of Thomas Penyngton Kirkman and the systematic development of finite projective geometry given by von Staudt The first axiomatic treatment of finite projective geometry was developed by the Italian mathematician Gino Fano.