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Pr ojecti planes Projecti and af ﬁne planes are more than just spaces of smallest (non-tri vial) dimension: as we will see, the are truly xceptional, and also the play crucial ole in the coordinatisation of arbitrary spaces. 2.1 Pr ojecti planes ha seen in Sections 1.2 and 1.3 that, for an ﬁeld the geometry PG has the follo wing properties: (PP1) An tw points lie on xactly one line. (PP2) An tw lines meet in xactly one point. (PP3) There xist four points, no three of which are collinear will no use the term pr ojective plane in more general sense, to refer to an structure of points and lines which satisﬁes conditions (PP1)-(PP3) abo e. In projecti plane, let and be point and line which are not incident. The incidence deﬁnes bijection between the points on and the lines through By (PP3), gi en an tw lines, there is point incident with neither; so the tw lines contain equally man points. Similarly each point lies on the same number of lines; and these tw constants are equal. The or der of the plane is deﬁned to be one less than this number The order of PG is equal to the cardinality of (W sa in the last section that projecti line er GF has points; so PG is projecti plane of order In the inﬁnite case, the claim follo ws by simple cardinal arithmetic.) 19

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20 2. Pr ojective planes Gi en ﬁnite projecti plane of order each of the lines through point contains further points, with no duplications, and all points are accounted for in this ay So there are points, and the same number of lines. The points and lines form 2- design. The con erse is also true (see Ex ercise 2). Do there xist projecti planes not of the form PG The easiest such xamples are inﬁnite; gi tw completely dif ferent ones belo Finite xamples will appear later Example 1: ee planes. Start with an conﬁguration of points and lines ha ving the property that tw points lie on at most one line (and dually), and satisfying (PP3). Perform the follo wing construction. At odd-numbered stages, introduce ne line incident with each pair of points not already incident with line. At en-numbered stages, act dually: add ne point incident with each pair of lines for which such point doesn yet xist. After countably man stages, projecti plane is obtained. or gi en an tw points, there will be an earlier stage at which both are introduced; by the ne xt stage, unique line is incident with both; and no further line incident with both is added subsequently; so (PP1) holds. Dually (PP2) holds. Finally (PP3) is true initially and remains so. If we start with conﬁguration violating Desar gues Theorem (for xample, the Desar gues conﬁguration with the line pqr “brok en into separate lines pq qr ), then the resulting plane doesn satisfy Desar gues Theorem, and so is not PG Example 2: Moulton planes. ak the ordinary real af ﬁne plane. Imagine that the lo wer half-plane is refracting medium which bends lines of positi slope so that the part belo the axis has twice the slope of the part abo e, while lines with ne gati (or zero or inﬁnite) slope are unaf fected. This is an af ﬁne plane, and has unique completion to projecti plane (see later). The resulting projecti plane ails Desar gues theorem. see this, dra Desar gues conﬁguration in the ordinary plane in such ay that just one of its ten points lies belo the axis, and just one line through this point has positi slope. The ﬁrst xamples of ﬁnite planes in which Desar gues Theorem ails were constructed by eblen and edderb urn [38]. Man others ha been found since, ut all kno wn xamples ha prime po wer order The Bruc k–Ryser Theor em [4] asserts that, if projecti plane of order xists, where or (mod 4), then must be the sum of tw squares. Thus, for xample, there is no projecti plane of order or 14. This theorem gi es no information about 10, 12, 15, 18,

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2.1. Pr ojective planes 21 Recently Lam, Swiercz and Thiel [21 sho wed by an xtensi computation that there is no projecti plane of order 10. The other alues mentioned are undecided. An af ﬁne plane is an incidence structure of points and lines satisfying the follo wing conditions (in which tw lines are called par allel if the are equal or disjoint): (AP1) points lie on unique line. (AP2) Gi en point and line there is unique line which contains and is parallel to (AP3) There xist three non-collinear points. Remark. Axiom (AP2) for the real plane is an equi alent form of Euclid “par allel postulate”. It is called “Playf air Axiom”, although it as stated xplicitly by Proclus. Again it holds that is an af ﬁne plane. More generally if line and all its points are remo ed from projecti plane, the result is an af ﬁne plane. (The remo ed points and line are said to be “at inﬁnity”. lines are parallel if and only if the contain the same point at inﬁnity. Con ersely let an af ﬁne plane be gi en, with point set and line set It follo ws from (AP2) that parallelism is an equi alence relation on Let be the set of equi alence classes. or each line let where is the parallel class containing Then the structure with point set and line set is projecti plane. Choosing as the line at inﬁnity we reco er the original af ﬁne plane. will ha more to say about af ﬁne planes in Section 3.5. Exer cises 1. Sho that structure which satisﬁes (PP1) and (PP2) ut not (PP3) must be of one of the follo wing types: (a) There is line incident with all points. An further line is singleton, repeated an arbitrary number of times. (b) There is line incident with all points xcept one. The remaining lines all contain tw points, the omitted point and one of the others. 2. Sho that 2- design (with 1) is projecti plane of order 3. Sho that, in ﬁnite af ﬁne plane, there is an inte ger such that

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22 2. Pr ojective planes ery line has points; ery point lies on lines; there are points; there are parallel classes with lines in each. (The number is the or der of the af ﬁne plane.) 4. (The riendship Theor em .) In ﬁnite society an tw indi viduals ha unique common friend. Pro that there xists someone who is eryone else friend. [Let be the set of indi viduals, where is the set of friends of Pro that, in an countere xample to the theorem, is projecti plane, of order say No let be the real matrix of order 1, with entry if and are friends, otherwise. Pro that nI where is the identity matrix and the all-1 matrix. Hence sho that the real symmetric matrix has eigen alues (with multiplicity 1) and Using the act that has trace 0, calculate the multiplicity of the eigen alue and hence sho that 1.] 5. Sho that an Desar gues conﬁguration in free projecti plane must lie within the starting conﬁguration. [Hint: Suppose not, and consider the last point or line to be added.] 2.2 Desar guesian and appian planes It is no coincidence that we distinguished the free and Moulton planes from PG in the last section by the ailure of Desar gues Theorem. Theor em 2.1 pr ojective plane is isomorphic to PG for some if and only if it satisﬁes Desar gues Theor em. do not propose to gi detailed proof of this important result; ut some comments on the proof are in order sa in Section 1.3 that, in PG the ﬁeld operations (addition and mul- tiplication) can be deﬁned geometrically once set of four points with no three

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2.2. Desar guesian and appian planes 23 collinear has been chosen. By (PP3), such set of points xists in an projecti plane. So it is possible to deﬁne tw binary operations on set consisting of line with point remo ed, and to coordinatise the plane with this algebraic ob- ject. No it is ob vious that an ﬁeld axiom translates into certain “conﬁguration theorem”, so that the plane is PG if and only if all these “conﬁguration theorems hold. What is not ob vious, and quite remarkable, is that all these “con- ﬁguration theorems follo from Desar gues Theorem. Another method, more dif ﬁcult in principle ut much easier in detail, xploits the relation between Desar gues Theorem and collineations. Let be point and line. centr al collineation with centre and axis is collineation ﬁxing ery point on and ery line through It is called an elation if is on homolo gy otherwise. The central collineations with centre and axis form group. The plane is said to be tr ansitive if this group permutes transiti ely the set for an line on (or equi alently the set of lines on dif ferent from and pq where is point of ). Figure 2.1: The Desar gues conﬁguration Theor em 2.2 pr ojective plane satisﬁes Desar gues Theor em if and only if it is -tr ansitive for all points and lines L.

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24 2. Pr ojective planes Pr oof Let us tak another look at the Desar gues conﬁguration (Fig. 2.1). It is clear that an central conﬁguration with centre and axis which carries to is completely determined at ery point not on (The line meets at ﬁx ed point and is mapped to so is the intersection of and ob .) No if we replace with another line through we get another determination of the action of the collineation. It is easy to see that the condition that these tw speciﬁcations agree is precisely Desar gues Theorem. The proof sho ws little more. Once the action of the central collineation on one point of + is kno wn, the collineation is completely determined. So, if Desar gues Theorem holds, then these groups of central collineations act sharply transiti ely on the rele ant set. No the additi and multiplicati structures of the ﬁeld turn up as groups of elations and homologies respecti ely with ﬁx ed centre and axis. see im- mediately that these structures are both groups. More of the axioms are easily deduced too. or xample, let be line, and consider all elations with axis (and arbitrary centre on ). This set is group or each point on the elations with centre form normal subgroup. These normal subgroups partition the non-identity elements of since non-identity elation has at most one cen- tre. But group ha ving such partition is abelian (see Ex ercise 2). So addition is commutati e. In vie of this theorem, projecti planes er sk ﬁelds are called Desar guesian planes There is much more to be said about the relationships among conﬁguration theorems, coordinatisation, and central collineations. refer to Dembo wski book for some of these. One such relation is of particular importance. appus Theor em is the assertion that, if alternate ertices of he xagon are collinear (that is, the ﬁrst, third and ﬁfth, and also the second, fourth and sixth), then also the three points of intersection of opposite edges are collinear See Fig. 2.2. Theor em 2.3 pr ojective plane satisﬁes appus Theor em if and only if it is isomorphic to PG for some commutative ﬁeld Pr oof The proof in olv es tw steps. First, purely geometric ar gument sho ws that appus Theorem implies Desar gues’. This is sho wn in Fig. 2.3. This ﬁg- ure sho ws potential Desar gues conﬁguration, in which the required collinearity is sho wn by three applications of appus Theorem. The proof requires four ne

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2.2. Desar guesian and appian planes 25 Figure 2.2: appus Theorem points, os oa and os No ap- pus Theorem, applied to the he xagon osc sho ws that are collinear; applied to osb sho ws that are collinear; and applied to uvc (us- ing the tw collinearities just established), sho ws that are collinear The deri ed collinearities are sho wn as dotted lines in the ﬁgure. (Note that the ﬁgure sho ws only the generic case of Desar gues Theorem; it is necessary to tak care of the possible de generacies as well.) The second step in olv es the use of coordinates to sho that, in Desar guesian plane, appus Theorem is equi alent to the commutati vity of multiplication. (See Ex ercise 3.) In vie of this, projecti planes er commutati ﬁelds are called appian planes Remark. It follo ws from Theorems 2.1 and 2.3 and edderb urn Theorem 1.1 that, in ﬁnite projecti plane, Desar gues Theorem implies appus’. No geo- metric proof of this implication is kno wn. similar treatment of af ﬁne planes is possible. Exer cises 1. (a) Sho that collineation which has centre has an axis, and vice ver sa

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26 2. Pr ojective planes Figure 2.3: appus implies Desar gues (b) Sho that collineation cannot ha more than one centre. 2. The group has amily of proper normal subgroups which partition the non-identity elements of Pro that is abelian. 3. In PG let the ertices of he xagon be and < Sho that alternate ertices lie on the lines deﬁned by the column ectors >= and >? @= Sho that opposite sides meet in the points >? ba and A Sho that the second and third of these lie on the line B? ba which also contains the ﬁrst if and only if ab ba

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2.3. Pr ojectivities 27 2.3 Pr ojecti vities Let C be projecti plane. emporarily let be the set of points incident with and let be the set of lines incident with If is not incident with there is natural bijection between and each point on lies on unique line through This bijection is called per spectivity By iterating perspecti vities and their in erses, we get bijection (called pr ojectivity between an tw sets or In particular for an line we obtain set of projecti vities from to itself (or self-pr ojectivities ), and analogously set for an point The sets and are actually groups of permutations of or (An self-projecti vity is the composition of chain of perspecti vities; the product of tw self-projecti vities corresponds to the concatenation of the chains, while the in erse corresponds to the chain in re erse order .) Moreo er these permutation groups are naturally isomorphic: if is an projecti vity from to say then So the group of self-projecti vities on line is an in ariant of the projecti plane. It turns out that the structure of this group carries information about the plane which is closely related to concepts we ha already seen. Figure 2.4: 3-transiti vity Pr oposition 2.4 The permutation gr oup is -tr ansitive Pr oof It suf ﬁces to sho that there is projecti vity ﬁxing an tw points and mapping an further point to an other point In general, we will use

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28 2. Pr ojective planes the notation “( to via ) for the composite of the perspecti vities IH and JH Let be an other lines through 2), point on and 2) such that uz are collinear 2). Then the product of to via and to via is the required projecti vity (Fig. 2.4.) permutation group is sharply -tr ansitive if, gi en an tw -tuples of distinct points, there is unique element of carrying the ﬁrst to the second (in order). The main result about groups of projecti vities is the follo wing theorem: Figure 2.5: Composition of projecti vities Theor em 2.5 The gr oup of pr ojectivities on pr ojective plane is sharply -tr ansitive if and only if is pappian. Pr oof sk etch the proof. The crucial step is the equi alence of appus Theo- rem to the follo wing assertion: Let be non-concurrent lines, and and tw points such that the projecti vity to via to via ﬁx es Then there is point such that the projecti vity is equal to to via ).

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2.3. Pr ojectivities 29 The hypothesis is equi alent to the assertion that and are collinear No the point is determined, and appus Theorem is equi alent to the assertion that it maps random point of correctly (Fig. 2.5 is just appus Theorem.) No this assertion allo ws long chains of projecti vities to be shortened, so that their action can be controlled. The con erse can be seen another ay By Theorem 2.3, we kno that appian plane is isomorphic to PG for some commutati ﬁeld No it is easily check ed that an self-projecti vity on line is induced by linear fractional transformation (an element of PGL and this group is sharply 3-transiti e. In the ﬁnite case, there are ery fe 3-transiti groups apart from the sym- metric and alternating groups; and, for all kno wn non-P appian planes, the group of projecti vities is indeed symmetric or alternating (though it is not kno wn whether this is necessarily so). Both possibilities occur; so, at present, all that this pro vides us for non-P appian ﬁnite planes is single Boolean in ariant. In the inﬁnite case, ho we er more interesting possibilities arise. If the plane has order then the group of projecti vities has generators, and so has order so it can ne er be the symmetric group (which has order ). Barlotti [1] ga an xample in which the stabiliser of an six points is the identity and the stabiliser of an points is free group. On the other hand, Schleiermacher [25] sho wed that, if the stabiliser of an points is tri vial, then the stabiliser of an three points is tri vial (and the plane is appian). Further de elopments in olv deeper relationships between projecti vities, con- ﬁguration theorems, and central collineations; the deﬁnition and study of projec- ti vities in other incidence structures; and so on.

21 Pr ojecti planes ha seen in Sections 12 and 13 that for an 64257eld the geometry PG has the follo wing properties PP1 An tw points lie on xactly one line PP2 An tw lines meet in xactly one point PP3 There xist four points no three of which are co ID: 23050

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Pr ojecti planes Projecti and af ﬁne planes are more than just spaces of smallest (non-tri vial) dimension: as we will see, the are truly xceptional, and also the play crucial ole in the coordinatisation of arbitrary spaces. 2.1 Pr ojecti planes ha seen in Sections 1.2 and 1.3 that, for an ﬁeld the geometry PG has the follo wing properties: (PP1) An tw points lie on xactly one line. (PP2) An tw lines meet in xactly one point. (PP3) There xist four points, no three of which are collinear will no use the term pr ojective plane in more general sense, to refer to an structure of points and lines which satisﬁes conditions (PP1)-(PP3) abo e. In projecti plane, let and be point and line which are not incident. The incidence deﬁnes bijection between the points on and the lines through By (PP3), gi en an tw lines, there is point incident with neither; so the tw lines contain equally man points. Similarly each point lies on the same number of lines; and these tw constants are equal. The or der of the plane is deﬁned to be one less than this number The order of PG is equal to the cardinality of (W sa in the last section that projecti line er GF has points; so PG is projecti plane of order In the inﬁnite case, the claim follo ws by simple cardinal arithmetic.) 19

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20 2. Pr ojective planes Gi en ﬁnite projecti plane of order each of the lines through point contains further points, with no duplications, and all points are accounted for in this ay So there are points, and the same number of lines. The points and lines form 2- design. The con erse is also true (see Ex ercise 2). Do there xist projecti planes not of the form PG The easiest such xamples are inﬁnite; gi tw completely dif ferent ones belo Finite xamples will appear later Example 1: ee planes. Start with an conﬁguration of points and lines ha ving the property that tw points lie on at most one line (and dually), and satisfying (PP3). Perform the follo wing construction. At odd-numbered stages, introduce ne line incident with each pair of points not already incident with line. At en-numbered stages, act dually: add ne point incident with each pair of lines for which such point doesn yet xist. After countably man stages, projecti plane is obtained. or gi en an tw points, there will be an earlier stage at which both are introduced; by the ne xt stage, unique line is incident with both; and no further line incident with both is added subsequently; so (PP1) holds. Dually (PP2) holds. Finally (PP3) is true initially and remains so. If we start with conﬁguration violating Desar gues Theorem (for xample, the Desar gues conﬁguration with the line pqr “brok en into separate lines pq qr ), then the resulting plane doesn satisfy Desar gues Theorem, and so is not PG Example 2: Moulton planes. ak the ordinary real af ﬁne plane. Imagine that the lo wer half-plane is refracting medium which bends lines of positi slope so that the part belo the axis has twice the slope of the part abo e, while lines with ne gati (or zero or inﬁnite) slope are unaf fected. This is an af ﬁne plane, and has unique completion to projecti plane (see later). The resulting projecti plane ails Desar gues theorem. see this, dra Desar gues conﬁguration in the ordinary plane in such ay that just one of its ten points lies belo the axis, and just one line through this point has positi slope. The ﬁrst xamples of ﬁnite planes in which Desar gues Theorem ails were constructed by eblen and edderb urn [38]. Man others ha been found since, ut all kno wn xamples ha prime po wer order The Bruc k–Ryser Theor em [4] asserts that, if projecti plane of order xists, where or (mod 4), then must be the sum of tw squares. Thus, for xample, there is no projecti plane of order or 14. This theorem gi es no information about 10, 12, 15, 18,

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2.1. Pr ojective planes 21 Recently Lam, Swiercz and Thiel [21 sho wed by an xtensi computation that there is no projecti plane of order 10. The other alues mentioned are undecided. An af ﬁne plane is an incidence structure of points and lines satisfying the follo wing conditions (in which tw lines are called par allel if the are equal or disjoint): (AP1) points lie on unique line. (AP2) Gi en point and line there is unique line which contains and is parallel to (AP3) There xist three non-collinear points. Remark. Axiom (AP2) for the real plane is an equi alent form of Euclid “par allel postulate”. It is called “Playf air Axiom”, although it as stated xplicitly by Proclus. Again it holds that is an af ﬁne plane. More generally if line and all its points are remo ed from projecti plane, the result is an af ﬁne plane. (The remo ed points and line are said to be “at inﬁnity”. lines are parallel if and only if the contain the same point at inﬁnity. Con ersely let an af ﬁne plane be gi en, with point set and line set It follo ws from (AP2) that parallelism is an equi alence relation on Let be the set of equi alence classes. or each line let where is the parallel class containing Then the structure with point set and line set is projecti plane. Choosing as the line at inﬁnity we reco er the original af ﬁne plane. will ha more to say about af ﬁne planes in Section 3.5. Exer cises 1. Sho that structure which satisﬁes (PP1) and (PP2) ut not (PP3) must be of one of the follo wing types: (a) There is line incident with all points. An further line is singleton, repeated an arbitrary number of times. (b) There is line incident with all points xcept one. The remaining lines all contain tw points, the omitted point and one of the others. 2. Sho that 2- design (with 1) is projecti plane of order 3. Sho that, in ﬁnite af ﬁne plane, there is an inte ger such that

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22 2. Pr ojective planes ery line has points; ery point lies on lines; there are points; there are parallel classes with lines in each. (The number is the or der of the af ﬁne plane.) 4. (The riendship Theor em .) In ﬁnite society an tw indi viduals ha unique common friend. Pro that there xists someone who is eryone else friend. [Let be the set of indi viduals, where is the set of friends of Pro that, in an countere xample to the theorem, is projecti plane, of order say No let be the real matrix of order 1, with entry if and are friends, otherwise. Pro that nI where is the identity matrix and the all-1 matrix. Hence sho that the real symmetric matrix has eigen alues (with multiplicity 1) and Using the act that has trace 0, calculate the multiplicity of the eigen alue and hence sho that 1.] 5. Sho that an Desar gues conﬁguration in free projecti plane must lie within the starting conﬁguration. [Hint: Suppose not, and consider the last point or line to be added.] 2.2 Desar guesian and appian planes It is no coincidence that we distinguished the free and Moulton planes from PG in the last section by the ailure of Desar gues Theorem. Theor em 2.1 pr ojective plane is isomorphic to PG for some if and only if it satisﬁes Desar gues Theor em. do not propose to gi detailed proof of this important result; ut some comments on the proof are in order sa in Section 1.3 that, in PG the ﬁeld operations (addition and mul- tiplication) can be deﬁned geometrically once set of four points with no three

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2.2. Desar guesian and appian planes 23 collinear has been chosen. By (PP3), such set of points xists in an projecti plane. So it is possible to deﬁne tw binary operations on set consisting of line with point remo ed, and to coordinatise the plane with this algebraic ob- ject. No it is ob vious that an ﬁeld axiom translates into certain “conﬁguration theorem”, so that the plane is PG if and only if all these “conﬁguration theorems hold. What is not ob vious, and quite remarkable, is that all these “con- ﬁguration theorems follo from Desar gues Theorem. Another method, more dif ﬁcult in principle ut much easier in detail, xploits the relation between Desar gues Theorem and collineations. Let be point and line. centr al collineation with centre and axis is collineation ﬁxing ery point on and ery line through It is called an elation if is on homolo gy otherwise. The central collineations with centre and axis form group. The plane is said to be tr ansitive if this group permutes transiti ely the set for an line on (or equi alently the set of lines on dif ferent from and pq where is point of ). Figure 2.1: The Desar gues conﬁguration Theor em 2.2 pr ojective plane satisﬁes Desar gues Theor em if and only if it is -tr ansitive for all points and lines L.

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24 2. Pr ojective planes Pr oof Let us tak another look at the Desar gues conﬁguration (Fig. 2.1). It is clear that an central conﬁguration with centre and axis which carries to is completely determined at ery point not on (The line meets at ﬁx ed point and is mapped to so is the intersection of and ob .) No if we replace with another line through we get another determination of the action of the collineation. It is easy to see that the condition that these tw speciﬁcations agree is precisely Desar gues Theorem. The proof sho ws little more. Once the action of the central collineation on one point of + is kno wn, the collineation is completely determined. So, if Desar gues Theorem holds, then these groups of central collineations act sharply transiti ely on the rele ant set. No the additi and multiplicati structures of the ﬁeld turn up as groups of elations and homologies respecti ely with ﬁx ed centre and axis. see im- mediately that these structures are both groups. More of the axioms are easily deduced too. or xample, let be line, and consider all elations with axis (and arbitrary centre on ). This set is group or each point on the elations with centre form normal subgroup. These normal subgroups partition the non-identity elements of since non-identity elation has at most one cen- tre. But group ha ving such partition is abelian (see Ex ercise 2). So addition is commutati e. In vie of this theorem, projecti planes er sk ﬁelds are called Desar guesian planes There is much more to be said about the relationships among conﬁguration theorems, coordinatisation, and central collineations. refer to Dembo wski book for some of these. One such relation is of particular importance. appus Theor em is the assertion that, if alternate ertices of he xagon are collinear (that is, the ﬁrst, third and ﬁfth, and also the second, fourth and sixth), then also the three points of intersection of opposite edges are collinear See Fig. 2.2. Theor em 2.3 pr ojective plane satisﬁes appus Theor em if and only if it is isomorphic to PG for some commutative ﬁeld Pr oof The proof in olv es tw steps. First, purely geometric ar gument sho ws that appus Theorem implies Desar gues’. This is sho wn in Fig. 2.3. This ﬁg- ure sho ws potential Desar gues conﬁguration, in which the required collinearity is sho wn by three applications of appus Theorem. The proof requires four ne

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2.2. Desar guesian and appian planes 25 Figure 2.2: appus Theorem points, os oa and os No ap- pus Theorem, applied to the he xagon osc sho ws that are collinear; applied to osb sho ws that are collinear; and applied to uvc (us- ing the tw collinearities just established), sho ws that are collinear The deri ed collinearities are sho wn as dotted lines in the ﬁgure. (Note that the ﬁgure sho ws only the generic case of Desar gues Theorem; it is necessary to tak care of the possible de generacies as well.) The second step in olv es the use of coordinates to sho that, in Desar guesian plane, appus Theorem is equi alent to the commutati vity of multiplication. (See Ex ercise 3.) In vie of this, projecti planes er commutati ﬁelds are called appian planes Remark. It follo ws from Theorems 2.1 and 2.3 and edderb urn Theorem 1.1 that, in ﬁnite projecti plane, Desar gues Theorem implies appus’. No geo- metric proof of this implication is kno wn. similar treatment of af ﬁne planes is possible. Exer cises 1. (a) Sho that collineation which has centre has an axis, and vice ver sa

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26 2. Pr ojective planes Figure 2.3: appus implies Desar gues (b) Sho that collineation cannot ha more than one centre. 2. The group has amily of proper normal subgroups which partition the non-identity elements of Pro that is abelian. 3. In PG let the ertices of he xagon be and < Sho that alternate ertices lie on the lines deﬁned by the column ectors >= and >? @= Sho that opposite sides meet in the points >? ba and A Sho that the second and third of these lie on the line B? ba which also contains the ﬁrst if and only if ab ba

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2.3. Pr ojectivities 27 2.3 Pr ojecti vities Let C be projecti plane. emporarily let be the set of points incident with and let be the set of lines incident with If is not incident with there is natural bijection between and each point on lies on unique line through This bijection is called per spectivity By iterating perspecti vities and their in erses, we get bijection (called pr ojectivity between an tw sets or In particular for an line we obtain set of projecti vities from to itself (or self-pr ojectivities ), and analogously set for an point The sets and are actually groups of permutations of or (An self-projecti vity is the composition of chain of perspecti vities; the product of tw self-projecti vities corresponds to the concatenation of the chains, while the in erse corresponds to the chain in re erse order .) Moreo er these permutation groups are naturally isomorphic: if is an projecti vity from to say then So the group of self-projecti vities on line is an in ariant of the projecti plane. It turns out that the structure of this group carries information about the plane which is closely related to concepts we ha already seen. Figure 2.4: 3-transiti vity Pr oposition 2.4 The permutation gr oup is -tr ansitive Pr oof It suf ﬁces to sho that there is projecti vity ﬁxing an tw points and mapping an further point to an other point In general, we will use

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28 2. Pr ojective planes the notation “( to via ) for the composite of the perspecti vities IH and JH Let be an other lines through 2), point on and 2) such that uz are collinear 2). Then the product of to via and to via is the required projecti vity (Fig. 2.4.) permutation group is sharply -tr ansitive if, gi en an tw -tuples of distinct points, there is unique element of carrying the ﬁrst to the second (in order). The main result about groups of projecti vities is the follo wing theorem: Figure 2.5: Composition of projecti vities Theor em 2.5 The gr oup of pr ojectivities on pr ojective plane is sharply -tr ansitive if and only if is pappian. Pr oof sk etch the proof. The crucial step is the equi alence of appus Theo- rem to the follo wing assertion: Let be non-concurrent lines, and and tw points such that the projecti vity to via to via ﬁx es Then there is point such that the projecti vity is equal to to via ).

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2.3. Pr ojectivities 29 The hypothesis is equi alent to the assertion that and are collinear No the point is determined, and appus Theorem is equi alent to the assertion that it maps random point of correctly (Fig. 2.5 is just appus Theorem.) No this assertion allo ws long chains of projecti vities to be shortened, so that their action can be controlled. The con erse can be seen another ay By Theorem 2.3, we kno that appian plane is isomorphic to PG for some commutati ﬁeld No it is easily check ed that an self-projecti vity on line is induced by linear fractional transformation (an element of PGL and this group is sharply 3-transiti e. In the ﬁnite case, there are ery fe 3-transiti groups apart from the sym- metric and alternating groups; and, for all kno wn non-P appian planes, the group of projecti vities is indeed symmetric or alternating (though it is not kno wn whether this is necessarily so). Both possibilities occur; so, at present, all that this pro vides us for non-P appian ﬁnite planes is single Boolean in ariant. In the inﬁnite case, ho we er more interesting possibilities arise. If the plane has order then the group of projecti vities has generators, and so has order so it can ne er be the symmetric group (which has order ). Barlotti [1] ga an xample in which the stabiliser of an six points is the identity and the stabiliser of an points is free group. On the other hand, Schleiermacher [25] sho wed that, if the stabiliser of an points is tri vial, then the stabiliser of an three points is tri vial (and the plane is appian). Further de elopments in olv deeper relationships between projecti vities, con- ﬁguration theorems, and central collineations; the deﬁnition and study of projec- ti vities in other incidence structures; and so on.

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