Fano plane

The Fano plane is an incidence structure that can be conceived both as a linear space as a projective plane, two-dimensional projective space or a block diagram. It is named after the Italian mathematician Gino Fano. In the synthetic geometry it is the minimal model of a projective plane. Your affine neckline that is created by cutting a projective line, is the minimal model of an affine plane.

The automorphism group of the Fano plane is the group of their projectivities, symbolically represented, as it is formally a factor group of the general linear group, it is in fact isomorphic to this. is a simple group and one in the classification of finite simple groups to the smallest non-commutative simple groups. It ranks up there among the groups of Lie type.

In addition, be in the language of synthetic geometry those projective or (more rarely) affine planes called Fano planes in which the Fano axiom applies. The Fano plane, as this article describes is not a Fano plane in the axiomatic sense, because it does not satisfy the projective Fano axiom.

  • 2.1 degeneracies
  • 3.1 Straight coordinates and duality
  • 4.1 2 groups and Perspektivitäten
  • 4.2 3 groups, and the affinity group
  • 4.3 7 groups and small systems of generators
  • 5.1 General
  • 5.2 Applications of the Fano plane on math puzzles
  • 5.3 The group-theoretical statements

Definitions

Visualization, definition as a hypergraph

The Fano plane can be explained by drawing an equilateral triangle with ups and visualize inscribed circle (first picture above) and define. 7, the elements of the points, the three vertices, three Höhenfußpunkte and the center of the inscribed circle. The 7 elements of that line, then the sides of the triangle, the highs and the inscribed circle. This image can - in the sense of graph theory - as a hypergraph with nodes ( points ) and edges ( the routes and the inscribed circle ) and thus regarded as a model of the Fano plane.

Concrete, enumerative definition as the incidence structure

The following (balanced ) structure is known as the Fano incidence plane:

In this specific definition by listing the 7 " binary" point symbols (see the figure on the right :, etc.) are simply taken as idiosyncratic symbols for 7 different points. In fact, they are an abbreviation for coordinate triplets, as will be explained below. The second, more compact representation of points and straight quantity produced in each case by the interpretation of these point symbols as numbers in the binary system and their conversion to decimal.

Definition as a projective space

Equivalence can be the Fano plane define the language of linear algebra as the two-dimensional projective space over the finite field with two elements. This body, in turn, can be modeled by the residue class field.

Equivalence of the definitions

The Fano plane is in the sense of synthetic geometry is a finite projective plane of order 2 with 7 lines and 7 points, their symbolic abbreviation. The axiomatic description of projective planes, this follows by reference to the incidence structure defined concretely we immediately checked the validity of the axioms.

When defining a two-dimensional projective space over the field we consider the vector space whose one-dimensional subspaces are then the points of the projective plane, its two-dimensional subspaces of the lines and the incidence relation is the set-theoretical subset relation " ". Thus we obtain ( formally)

  • As a point set: and
  • As a set of lines.

Here are the icons for the generated by the vector or the vectors of linear subspaces. The condition in the definition of the set of lines is equal to the vector space of this linear independence of the two vectors, as long as the zero vector is excluded.

Point coordinates

Now you can in the visualized model ( equilateral triangle with heights and inscribed circle ) a complete quadrilateral, ie an ordered four-element point set in which no three points lie on a line, select a projective point basis and these points in the given order, the ( products of ) standard basis of the plus point of unity assign: etc. - formal or more precisely, because of the projective point corresponds to a one-dimensional subspace, which is the product of. Now, the notation is made ​​somewhat informal and shortened further: We agree as an abbreviation for the assignment described above.

In the second picture above, this was done. The corners were in the order " right 100, top 010, left 001" to the first three basis points and the orthocenter to the unit point, the coordinates of the remaining points are thus obtained: Must The third point on a line by binary addition without carry-overs (also exclusive-OR operation XOR called ) the other two points on the line result. For example: in words, the center of the lower side of the equilateral triangle ( 101) is located on the lower side (first " sum " ), the amount to this page (second " sum " ) and on the inner circle ( third " sum " ). That these equations for the rise "Buzz " now just means that the vector lies within the relevant product of the summed, different coordinate vectors. Since the three middles, so just the points that do not belong to the complete quadrangle can be koordinatisiert consistent with this rule, the " visualization as an equilateral triangle", is formally more precisely, the corresponding hypergraph isomorphic to. The definition by enumeration are now but simply the set of points and straight amount of hypergraphs again and is therefore to the other two models are isomorphic.

Properties

  • The Fano plane is a two-dimensional projective space over a finite field in the sense of linear algebra.
  • As such it is a linear space.
  • She is a two-dimensional projective geometry pappussche and therefore a projective plane in the sense of synthetic geometry.
  • Every projective plane of order 2 is isomorphic to the Fano plane and there is no projective plane of a lower order.
  • It is a symmetric ( 7,3,1 ) block plan so that it is the smallest Hadamard block diagram.

Degeneracies

Due to their small size, the Fano plane some special features:

  • It fulfills the set of Pappus were " empty": Since there are no non-degenerate hexagons the way it requires the Pappus configuration, fall (at least) two of the six corners together. But then also fall at least two " points of intersection of opposite sides " together and the statement of the theorem that the three intersection points lie on a line, is trivial.
  • Since a non-degenerate Desargues configuration requires ten different points, and the set of Desargues will eventually be satisfied trivially.
  • It is the only projective plane over a division ring in which every projective perspectivity absolutely is the identity with a center outside of an axis. Through the guidelines, the "free" sets of points on the straight line through the center of one-pointed (at least one point of each of these lines is on, another is the center ). So the Transitivitätsforderung at levels of Lenz - Barlotti class VII.2 is fulfilled empty. This class VII.2 therefore include the Fano plane like any other desarguesian level.
  • The cross ratio is degenerate, since there are no straight lines on four different points.
  • The Fano plane is the only projective plane in which the point of unity for the definition of a projective point base is not really necessary: In the vector space model, each one-dimensional space contains the zero vector only another point, that is, the indirect attribution " basis vector → one-dimensional subspace ↔ projective point "here is also the basis vector towards reversible without a point of unity must be taken to help; this is the way, exactly as for the projective spaces of any dimension if, but only if the body has exactly two elements. When choosing the point base in the plane, this special shows like this: If you have chosen three non- collinear points for point basis - for this choice there is when taking into account the order after all possibilities - then there is always exactly one point in the plane of the is not collinear with the selected two points!
  • The automorphism group of the latter peculiarity suppressed by the fact that from the valid for any body Isomorphieaussage follows the statement, because the multiplicative group is for the two-element body the fuel group.

Dualization

An abstract dualization of the Fano plane is obtained by interchanging the definitions in a point set and set of lines and the incidence relation is reversed, so consider the incidence structure. The so derived from incidence structure is always another incidence structure and a projective plane (in the sense of synthetic geometry) again a projective plane of the same order. For desarguesian projective planes and thus also for the Fano plane, the dual structure to the original structure is isomorphic. This one shows with the help of a concrete isomorphism ( a correlation), which depends on the chosen coordinate system. Such a correlation ( here more precisely, a projective, hyperbolic polarity ) is described in the following section:

Just coordinate and duality

In the vector space model, each can be straight, ie describe each two-dimensional subspace of a homogeneous linear equation. The coordinate vector of the straight line (straight line coordinates) is so. A point with projective coordinates, the straight line with the homogeneous coordinates is precisely allocated and vice versa. The figure at right shows the mappings for the conclusion reached in this article choice of base point: The dots at the bottom of the image are assigned by this correlation the straights that are directly above them, the lines which are directly under them point. For example, the 3 basis point 001, the lower-right corner of the triangle, the line with the equation ( the left side of the triangle ) is assigned, the height through the top tip with the equation their Höhenfußpunkt 101

In the picture the lines represent in the middle between the red highlighted lines and dots, the incidence relation is to be reversed by the dualization: or set theory, in the vector space model, the less-than symbol for " is linear subspace of " is.

Collineation

The automorphism group of the Fano plane is the group of their collineations; it agrees with the group their projectivities, since the two-element prime field does not allow non-identical Körperautomorphismen. It operates sharply transitively on the ordered non-collinear Punktetripeln and therefore has the order, the same order is obtained from the formula for the order of the general linear group. It is nichtabelsch and simple ( ie, it has only the trivial normal subgroup ).

In the following, group-theoretical considerations, it is represented as a group of permutations of its points, so as a subgroup of the symmetric group. It is agreed that permutations operate from left to numbers, that is, it applies and, for example. In the language of geometry projectivities two are mutually conjugate if and only if has the same representation as in regard to our starting point ( as a linear map ) with respect to a suitable point basis. In the permutation means that the plane can be renumbered according to the above described " binary" system that the permutation of takes. Two permutations are exactly then conjugated if they have similar representations as products of disjoint cycles, this condition is therefore also necessary, it turns out there too - except for the elements of order 7, which fall into two conjugacy classes - as sufficiently.

2 groups and Perspektivitäten

  • For each of the 7 line, a group of flat Perspektivitäten exists with the axis. Has a - perspectivity a further checkpoint outside the axis, then it is the identity, since each has only three straight points. Therefore, all non-identical Perspektivitäten have its center on the axis and swap the two points that lie on each except by the two different axis of the straight line: The group of Perspektivitäten with axle and the center is a cyclic group with two elements from the isomorphism.

3 groups, and the affinity group

  • If the points are on a straight line, thus creating two of the Perspektivitätsgruppen together a subset of 24 elements. For example, as a product of Perspektivitäten the product of these three generators is the element, an element of order 3 This element is conjugate to its inverse in already, therefore contains an isomorphic to the dihedral subgroup. As a group with 24 elements contains three of the 2- Sylowgruppen the collineation group with 8 elements: for
  • The group is isomorphic to the symmetric group, as it is the Just when Fixgerade from up and operates a loyalty to the longitudinally split level as a group of affinities. The full affinity group of the affine plane with 4 points, however, is just that.
  • All 56 elements of order 3 are conjugate to each other and so they always have the form of a product of two disjoint 3- cycles, with one cycle contains collinear points, the other three of the four non-collinear points outside the order determined by the first cycle straight line.
  • Each of 28 3- Sylowgruppen the group order 3 is generated by one of said elements of order 3.

7 groups and small systems of generators

  • In exist exactly 48 projectivities of order 7, which can be represented respectively as a 7- cycle.
  • Each of these group elements creates a 7- Sylowgruppe of the isomorphism, which contains a total of six 7 -cycles. Two different this 7- Sylowgruppen intersect in the trivial group, so there are exactly 8 sub-groups of this type.
  • The amount of 7 -cycles is divided into into two conjugacy classes, each with 24 elements.
  • Each 7- cycle is completely determined by any four points which he depicts another in this order, which are not collinear and always
  • If an element of order 7 (7 cycle), then iff is conjugate to when a rest is square modulo 7, ie lie in the same conjugacy class as in the other.
  • The cyclic collineation group generated by a 7- cycle is a Singer cycle, therefore exists for each 7 - cycle is a bijection ( " renumbering " ) of the point set, according to the derived renumbered the Fano plane of the difference set.
  • If two projectivities of order 7, then applies after the Sylowsätzen and is as simple or. That is, the automorphism is produced by two seven -cycles suitably selected, for example, with
  • Each 7- cycle can be represented as a composition of exactly three different, non-identical Perspektivitäten. Is or with the Perspektivitäten For the two cycles from the previous statement.
  • It follows that as a product of three Perspektivitäten and as a product of a perspectivity and a projectivity of order 4
Linear space (geometry) Affine plane (incidence geometry) Null vector Correlation (projective geometry) Alternating group Klein four-group Translation plane Dihedral group Jack van Lint Wolfram Research
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