1.

COLLINEATION GROUPS OF PROJECTIVE. PLANES AND SPACES ...... 193221==1.1===11111

by

FREDERICK CHARLES PIPER

A Thesis submitted for the

degree of Doctor of Philosophy

in the University of London.

IMPERIAL COLLEGE November, 1964 2

ABSTRACT

We begin this thesis by answering the following questions. If EP is a finite projective plane and if-frs -i a collineation group of c3 such that - (i) FT- contains some elations of QP (ii)a line of P is the axis of elations inl-F- for more than one centre (iii)-rT - does not fix a point or line of then (a) what can be said about -FT ? (b) what configurations may be formed by the centres and the axes of these elations? We then use our answers to prove that if every flag of q) is fixed by an elation in-1T, then R is desarguesian, and either (i)11- contains the little projective group of @ , or, (ii) 3 6 the order of (3 is four andiT A or s6° In § 5 we show that if every point of is the centre of a homology in "TT and if no point or line of P is fixed by -TT- then (P is desarguesian, and Tr- contains the little projective group of CP . It is then shown that if the order of CP is not a square, then the same conclusions hold if we assume that every point is the centre of a perspectivity in IT Finally, in § 6, some of the results about elations in planes are extended to spaces of higher dimension. We prove the following;

"Let Hk be a finite projective space of odd order and of 3

dimension k, whore k>21 and letTIbe a collineation group of Hi with the following properties contains some elations of Hk (ii) a hyparplane of Hk is the axis of elations in Tr for more than_ one centre (iii) no subspace of Hk is left fixed byll— thenTr contains the little projective group of Hk.H Also submitted as part of the thesis is a published note about elations in finite projective planes. 4.

ACKNOWLEDGMEN'IS

While preparing this Thesis I was helped by talks

with Dr. P. Dembowski, Dr. D. H. Hughes and

Dr. P. J. Higgins. I wish to thank them for

their assistance, and to thank Dr. Higgins for

acting as my supervisor while Dr. Wagner waa in

America.

Special thanks are due to my supervisor,

Dr. A. Wagner, for the many hours spent in discussions, and for his endless supply of ideas, encouragement and criticisms. 5

CONTENTS

1. Definitions and basic concepts 00000 00 .. 00001000 6.

2. Intr oduction .. 0 0 00 000 o00 trao0c000000.00600 a0000 15.

3. Preliminary results ...... 000000000 00000• 22.

4. On elations of finite projective planes . 0 27.

5. Perspectivities of finite projective planes 55.

6. Elation of finite projective spaces of odd order 65.

7. References •00•000000000 . 000000 00000000 000 00000. 72.

8. Notation and definition of 'connected set' 75.

X4RATA (see pages 68 & 69)

Lemma 22 should be read before Lemma 21.

The first two lines of the proof of Lemma 23. must be deleted, and should be replaced by

"Let P and R be any two centres. By lemma 22, P has an axis, Hki say, such that 14 Hic.a. Let Sk bei an axis of R, let Q be any other centre of Rk.-i and let e- PQ."

The rest of the proof of lemma 21 is correct. 6.

1. DEFINITIONS AND BASIC CONCEPTS

1.1 Projective Planes

We define a projective plane as a set 6), the elements of which we call points, together with. a system of subsets of 6', which we call lines, satisfying the following three axioms:-

1. Given any two distinct points, there exists a unique line containing these points.

2. Given any two distinct lines, there exists a unique point contained in both these lines,

3. There exist four points such that no three of them are contained in one line.

Throughout this thesis we shall use the following notation. Gothic upper case letters, e.g. CD , will be used for planes, sets of points and sets of lines. Roman upper case letters will be used for points and Roman lower case letters will be used for lines. If P and Q are any two distinct points and if e is the unique line containing them, we shall write e = K. If t.„1") and m are any two distinct lines and if 0 is the unique point contained in both of them, we shall write 0 = _e Ara. If P is any point and if (iis any line, we shall express that fact that P is contained in e_ by writing PE-2. If P is not contained in Ci , we shall write P,i..e.or, alternatively, PE 6) Q. If a projective plane contains only a finite number of points 7. the plane is said to be finite (Finite planes were first studied by Veblen and Bussey NI). It is easily proved that, for any finite projective plane 6), there exists a positive integer n such that

CO every line of U' contains n + 1 points and (Li.) every point of 6P is on n 1 lines. This number is called the order of the plane. Simple counting shows that a finite projective plane of order n contains 2%1.n-a points and ii+n+1 lines.

Any subset,, of the points of 6) which satisfies the axioms of a projective plane is said to be a subplane of 6). If every /n/ point of tr is on at least one lino of P, then O'is called a square-root subplane. This name arises from the fact that if 6? is, a finite projective plane of order n and if Oi ls a square-root subplane of 9, then n is a square and the order of V is If Pis a finite projective plane of oven (odd) order n, then a sot of n+2, (11-1), points such that no three are collinear is called an oval. (Ovals have bean extensively studied by s°gre [20, ?3)., An incident point-line pair is called a flag. Mappings of projective planes on to themselves play an important part in the theory of projective planes. Of particularimportance are the collineations. We define a collineation of a projective plane Pas a one-to-one mapping of the points of eon to the points of(dand of the linos ofe on. to the lines of Fsuch that incidence is preserved i.e. such that a point is on a line if, and only if, the image point is on the image lino.

The mapping which results from the carrying out of two successive collineations is again a collineation. Defining the product of two collinoations as the collineation obtained by 8. successively performing them, it is readily seen that the set of all collineations of a projective plane form a multiplicative group. We call this group the full collinoation groat of the plane. Any subset of collineations which form a subgroup of the full collineation group is called a collineation group. Throughout this thesis we shall use Greek upper case letters to denote groups, particularly collineation groups, and. Greek lower case letters to denote elements of groups, particularly collineations. The one exception being that we shall denote the identity element of any group by I. The image of a point P under a collineation e4. will be written as Pa._ . Consequently Q0 is the

collineation obtained by first performing 0: and then . cx, If a collinoation leaves fixed every point of some line, A -say, then 0( is. called a perspectivity. Baer DJ has shown that, if oCS, of...must also fix all the lines through some point, P say, and that can fix no other points or lines. The line e. is called the axis of the perspectivity and the point P is called its centre. We express the fact that P is the centre of 0( and that

is its axis by calling c3( a (P,e)-perspectivity. If P£'€-WO call the perspectivity an. elation, and if it is called a homology. Unless indications are given to the contrary, we shall use the term elation (homology) to mean an elation (homology) other than the identity. The collinoation group generated by all the Elation

of a desargues.ian plane 9 is called the little Projective gro..i.pt of Perspectivities assume a major role in the theory of projective planes. Their importance is due to their connection with the 9. Desargues configuration; a connection which was systematically studied by Baer [43.

Two triangles ABC, A/ BI C/are said to be in perspective from 0 if OA = 0/11 , OB = OB' and OC = OCR . They are said to be in

perspective from e if e /1AB = e Apb," BC = e A 131 CI and ACA = -e, /A/ It has been shown by Ostrom D13 that every, finite projective plane contains a. pair of triangles which are in. perspective from both a point and a line.

A projective plane 63 will be called (P e)-desarguesian if, for every two triangles in perspective from the fixed point P and such

that two pairs of corresponding sides intersect on the, fixed line 'e./.1 the intersection of the third pair of corresponding sides also lies on. e./. A projective plane will be said to be IL' t)- transitive if, for the fixed point P and the fixod line e-and any pair of points A, B where .A.,Bt Q, AfP, B P, PA=PB, there exists a (P, )-perspectivity 01C;Ith A etC =B. Baer CC has shown that a projective plane is (P, .)-desarguesian if, and only if, it is (P,,)-transitive. A projective plane is called desaresian if it is (P,2-)-desarguesian for all points P and all lines Hilbert [DJ showed that any desarguesian plane may by coordinatized by elements of a field. Consequently any collineation of a desarguesian plane may be represented by a semi-linear transformation of a vector space of dimension three over . The little projective group of a desarguesian plane is then isomorphic to the quotient group of the group of unimodular linear transformations of the vector space by its centre. 10. A projective plane is called alternative if it is (P, desarguesian for all points P and all lines with PE (Alternative planes were first studied by Miss Moufang E. .Any alternative plane may by coordinatized by the elements of an alternative field. Thus, since the Artin-Zorn, theorem states that every finite alternative field is a field, (for a proof see Levi [163 ), every finite alternative plane is dosarguesian. Examples do exist, however, of finite non-desarguesian planes. (See, for example, Hughes [10 and Ostrom [22]). All known finite projective planes fall into one of the following two classes: (1) the translation planes, (2) the semi-translation planes. We now define these two classes. I,. projective plane is called a translation plane if it is (P, e)-desarguesian for all points PE Q. where e-is some fixed line. A projective plane 6) of order 92: (where *is a prime power greater than two) is called a semi-translation plane with respect to the Ling e,- if there exis-ts a set an of cfr+ 1 points on. a, such -that, for any point PE(, 9 admits a group of elations of order crf with centre P and axis 'eF. If the total group of elation's with axis has order c , then 9 is called a strict semi-translation plane with respect to • The system Q obtained from a projective plane e by regarding as special all the points of a line (usually denoted by e„,„ ), is called an affine plane. The line eco is often called the special line of CY . The lines of kr other than are called the affine lines, and the points of lr which are not incident with ee,„ are called the affine points. The order of a finite affino plane is defined as the number of points on a line. It is equal to the 11. order of the finite projective plane from which it was derived. !n elation of with £ as axis is called a translation . The group generated by all translations of is called its translation group. Another type of mapping which is important in the theory of projective planes is the correlation. We define a correlation of a projective plane as a ono-to-one mapping of the points of e on to the lines of Q and of the lines of (53 on to the points of e such that incidence is preserved, i.e. a point is on a line if, and only if, the imago lino passes through the image point. It is clear that the mapping which results from successively carrying out two correlations is a collineation. A correlation 19 will be called a polarity if 91 = I, the identity collineation. A point

Q will be called an absolute point of the polarity 09 if Q£ Finally a collineation of order two is an involution. 12.

1.2 Projective Spaces

TIe define a projective space as a sot, (-I, 9 the elements of which we call points, together with a system of subsets of Ht.- 2 which we call lines, satisfying the following four axioms:

1. Given any two distinct points, there exists a unique line containing them, (We say the points, are on the line.)

2. If A, B, C are points not all on the same line, and if D

and E E) are points such that B, C, D are on. a lino and C, A, E are on a line, then there is a point F such that ...A,- E, F are on a. -line and also D, E,. F are on aline.

B. There are -at Least three -points • onn-a line

4. There are four points such that no three are on a line.

Many text books have been written about the theory of projective spaces (see, for example Veblen and Young Di] .In this section wo.just indicate these definitions which are relevant to .6.- If a projective space contains only a. finite number of points, the space is said to be finite. • It is easily proved that,. for any finite projective space , there exists a positive integer n such that each line of H e_ contains n 1 points. We call 31 the order of H e . Any subset Hs of the points and lines of He. which satisfy axioms 1-4 is called a subspace of HE.. If every point of Ht which is on a line of Hs is contained in Hs , then. Hs is called a complete subspace of He . Thus, if H e is a finite projective la.

space of order n and if Hs is a complete subspace of He. then the order of Hs is also n.

A set of points is said to sprIn a complete subspace Hs if

H s is the smallest complete subspace containing every point of the set. If the smallest number of points which span

H is then. we say that H. has dimension k. Points have dimension 0, lines have dimension 1, and planes have dimension 2.

A complete subspace of dimension k-1 is called a hyperplane.

Throughout the sections of this thesis which are concerned with projective spaces we shall use the following notation:

Roman upper case letters will be used for points and Roman lower case letters will be used for lines. Projective spaces will be represented by Roman upper case letters with Roman lower case letters indicating the dimension.

We define a collineation of a projective space HK as a one-to-one mapping of points on to points and lines on to lines such that incidence is. preserved. It is readily see that a collineation is then a permutation of the complete subspaces of

H k which preserves their dimensions and incidences.

A collineation o(.. of lit, which fixes all the points of a hyperplane H w=1.1-1011-• is called a RgyaRectivity. of Hp., 4, If a *1104, will also fix all the: h.yperplanes through some point P, and will fix no other points or hyperplanes. P is called the centre of o( , and 1 is its axis. If P £ I4t,_t we call cc_ an elation, and if P g vie call a( a homology.

The group generated by all the perspectivies of H . is called 14. the full projective group of H. . The group generated by all the elations of H e is called the little projective group of lib my projective. space of dimension greater than two is desarguesian and may be coordinatized by the elements of a. field F.

Any collineation of the space may then be represented as a semi- linear transformation of a kid dimensional vector space V over F.

The full projective group of H t is then isomorphic to the quotient group of the group of non-singular transformations of V by its centre, and the little projective group of H e is isomorphic to the quotient group of the group of unimodular linear transformations of V by its centre.

Any plane which may be embedded in a projective space of higher dimension is desarguesion. When We discuss projective planes in

4 and 5, we do not assume that they are desarguesian.

Consequently, throughout this thesis, vie take great care to distinguish between projective planes, and projective subspaces of dimension two. 15,

2 INTRODUCTION

2.1 Projective Planes

Since projective planes defined only by the axioms of incidence are of such a general nature, it is necessary to impose further

conditions on a plane before any informative investigation can be carried out.

Throughout this thesis we restrict our attention to planes of

finite order. By imposing no extra condition on their planes other than finiteness, Bruck and Ryser DJ have proved the non-existence of

certain infinite classes of finite projective planes. Usually, however, yet another assumption is made. The most commonly made

assumptions have been the existence of certain geometrical

configurations, or the existence of certain algebraic coordinatizing systems, or the existence of certain groups of collineations.

Our assumption is of the last typo. We are mainly interested in

collineation groups which contain perspectivitios of finite projective planes. We ask two questions. What do these

collineation groups look like? What configurations must be formed by the centres and axes of the perspectivitics?

Is was noticed in 1, the importance of porspectivities is duo to their connection with the Desargucs configuration. In fact, we pointed out that Baer showed that a plane is. (P,e.)-dosarguesian if, and only if, it is (P,(;)-transitive. From this it follows immediately that if a plane is (P,a)-transitive for every point P and every line ', then the plane is desarguesian. Since a finite alternative projective plane is desarguesian, a weaker condition for a finite plane to be desarguesian is that it is (Ple.)-trarmitive 16. far all points P and all lines -e with PE t. Two natural questions arise. Firstly, if a plane is (P, -ex )-transitive for

some points P and some lines , what configurations may be formed by these points and lines? Secondly, how many perspectivities are necessary to ensure that a plane is dosarguesian? The first question. has been. systematically studied by Lanz 0.5]

and Barlotti [6J The second question was considered by Lndre C13 and Gleason They proved the following:- "Let be a finite projective plane and lot Trbe a collineation group of Q. If for every, point P and every line , with P £ -e(P, e.), there exists a. non-trivia (P, b)-elation (homology) belonging to Tr, then 63 is desarguesian andli contains the little projective group of P.It The case where -TT-contains the elations was. due to Gleason and the case whareiTcontains the homologies was duo to Andre. In each case the proof consisted of showing that Q was (P, transitive in forevery point P and every line P., with PE P.-. These conditions wore then weakened by Wagner D.23 who showed that the same conclusions held if one assumed only that every point P was the centre of an elation (homology) in 11 , that every line was the axis of an elation (homology) in f and thatTrdid not fix a point or line of In Theorems 9 and 11 we weaken the conditions still further. We show that the same conclusions hold assuming only that every point P is the centre of an elation. (homology) in I and that Tr does not fix a point or line of Yet another weakening of the conditions is possible. In Theorem 10 we prove that if every flag of 6) is left fixed by an 17. elation inin and if does. not fix a point or line of (9, then

(? is desarguesian and that either containsthe little - projective group of 0, or the order of ? is four and ri AorA6or S The two generalizations of Gleason's Theorem are published in Ud] This final weakening of the conditions is not possible for homologies. If we assume that every flag of 62 is left fixed by a

homology in (I andthat (1 doesnot fix a point or line of (P

then, although this may imply that CP is desarguesian., need not

contain the little projective group of R. is a counter example, Professor MacLaughlin has pointed out that in the unitary group of a finite desarguesian plane. every flag is left fixed by an

involutory homology.

!n obvious question now arises. Is it possible to weaken the

conditions of Theorem 10 still further? If every point of C? is left fixed by an elation in- TT- , then is necessarily desarguesian? The following counter example shows that the answer is. no, and that the conditions of Theorem 10 are the weakest possible. Ostrom C?i] gives an example of a non-desarguesian plane of order q (where q is a prime power), which admits a collineation group-TT-containing the little projective group of a subplane of order q and containing no other elations. It is readily seen that every point of the plane is fixed by an elation in

In Theorem 13 we prove that if 6) is a finite projective plane of order n, n not a square, and if-IF-is a collineation group of (3) fixing no point or line such that every point P is the centre of a perspectivity in 11 thene is desarguesian and containsthe little projective group of e 18. The next question to ask is:- What about collineation groups

containing elations of non-desarguesian planes? In 1911, Mitchell c73 completely classified all subgroups of the full projective group of a desarguesian plane of odd order. His results were later extended to desarguesian planes of even order by Hartley all] . As a result of these papers, we know every

collineation group which contains, perspectivities of a finite desarguesian plane, and we also know the configuration formed by

the centres and axes of these perspectivities. Both of these papers make extensive use of the fact that, since the planes are

desarguesian they have a field as their coordinatizing system and

that all elations have the same order. They also make great use of Dickson's CIO results about linear groups on a line.

To attempt a similar classification for non-desarguesian planes appears to be much too difficult. In § 4 we restrict our

attention to collineation groups which contain elations. This restriction, however, is still not enough to determine any useful results and we impose the following fUrther conditions:-

(i) The collineation group does not fix a centre or axis.

(ii) There is a point of the plane which is an elation centre for more than ono axis.

(iii)There is a line of the plane which is an elation axis for more than one centre. Conditions (ii) and (iii) ar e needed to guarantee the existence of a certain number of elations of the same prime order (see

Results 2 and 3). Our results strongly suggest that conditions (i) and (ii) imply condition (iii), and dually. (Certainly this is 19. true in the desarguesian case). However, we have been unable to prove this.

For a desarguosian plane (P of odd order, Mitchell showed that

if a collineation group Trcontained some elations and if

conditions (i) (iii) were satisfied, then left invariant a (1/ subplane, say, of T and that, restricted to cr , -Trcontained the little projective group of . In Theorem 4 we show that

the same result is true for non-desarguesian planes of odd order. Since this result agrees with the special case of the desarguesian planes it is clearly the best possible result that one could hope for. Although there are no known examples of finite projective

planes which. admit elation of different prime order, it has not been shown that this cannot happen. It is an immediate* consequence of the above result that if a collineation group of a finite projective plane of odd order satisfies conditions (i)-(iii), then

all its elations have the same prime ardor.

For desarguesian planes of oven order the situation is not quite

so straightforward. Hartley showed that if a collimation group

contains involutary elation's and if conditions (i)-(iii) are satisfied, then-TT-loaves a subplane, ay, invariant. When restricted to Q , Treither contains the little projective group of eor is isomorphic to A 6 . If the latter case occurs, then ,t x the order of Lr is four. In Theorems 5;6 and 1 we show that either the same is true for non-desarguesian planes or else the centres and axes of the elations lie in disjoint subplanes of order two. If / is restricted to any one of those subplanes thenT— contains its little projective group. We have been unable to determine whether 20. more than one such subplano can actually, occur.

The concept of a connected sot plays a crucial role in. § 4.

Since this is a new idea and since the terms associated with it are so useful, we redefine a connected set, (and all the associated terms), on the last page of the thesis. This is to enable the reader to refer to the definitions more easily. Liso listed on that page is some, of the loss well known notation of this thesis. 21. 2.2 Projective Spaces

Since any collinoation of a projective space of dimension k may be represented by a somi-linear transformation of a vector space of dimension kid over its coordinatizing field, the collineation groups of projective spaces have been the object of extensive research for many years. Many text books have been written about them, and we draw particular attention to Artin . In §6 we consider a finite projective space, Hr , of odd order and a collineation group,71 , of containing some elations.

We show that if (a)-Tdoes not fix a subspace of Hk and (b) a hyperplane of Hh is the axis of elations inTrfor more than one -rr centre, then li must contain the little projective group of Hk .

22.

3„ PRELIMINARY RESULTS

Listed below are the previously known results which we require

for the proofs of our theorems. We assume an elementary knowledge of the theory of finite groups and of the theory of finite fields.

Where known we sive a reference to the first published proof of each result.

1 is a permutation group on. a set S. If T is a subset of S such that, given eny A E T, there exists an element of of given order p fixing A and no other element of 5, then T is contained in an orbit of S

(This is a simple extension to lemma 1.?, Gleason U.9],

Gleason's proof essentially still applies).

2 G) is a finite projective plane and—rris a, collineation group of cS) . If .2 is the axis of an elation in I and P E Q., is the centre of an elation in Tr then I contains

a (p, e, (This is a re-statement of Theorem 1, Wagner )

3 Q is a finite projective plane andTris a collinoation group of If for two distinct points X,Y, (X Ee , YEh n- contains non-trivial (X, ) and (Y, Q-,)-elations then all elations inTrwith axis P, have the same prime order.

(See Gleason 503 ).

4 Notation 1 group of all (X, t )-elations _7 2- C, )(EP._ Xie- G) is a finite projective plane. If, for some line -e,_ 23. all the groups Xe have the same order h >d , then 2e_ is transitive on the points of &\•12-. (See G1 eas on. 6] ) .

5 G) is a finite projective plane and-Tis a collineation group of e If e is (P,OP)-transitive for all points PE& , PP), then (.? is desarguesian. (due to work by Kleinfeld and San Souci. ) (See San Souci gri] )

6 G) is a finite projective plane and 1 I is a collineation

group of 6) If, for every point P9 \ contains an elation with centre P, and if, for every line 11 contains an

elation with axis C , then is desarguesian an.d7contains the little projective group of (See Wagner [3.21. )

7 G2) is a finite projective plane. Let 0( be a (P, C-)-

perspectivity of e and let Tr be any collineation of • Then 704-11 is a (P Q-111-perspectivity. (This is a widely used result. It is easily proved - see for example, Pickert ).

8 e is a finite projective plane and-Tris a collineation group of e fixing neither a point or a line. If every point of e is. the centre of a homology in and if every line of is. the axis of a homology in7 then Q is desarguesien and contains the little projective group of .

(See Wagner Da). 24.

9 Let Q be a finite projective plane of order n and let 0( be a perspectivity of order s. Then (1)

(ii) Cc is an elation if, and only if, sln (This is easily proved. Ostrom C0J uses it in a weaker form).

10 Q- is a finite affine plane and 0 is a point of Q00 . If

for every point P C'eco , there is a non-trivial (P,'e..)-

translation in then I I contains a non-trivial (0, t.,0)-translation.

(due to Hughes. For a proof see Piper

11 Let (be a finite affine plane and let o(J. ) 4 be two

homologies of with. centres CI and Ca_ respectively and

with common axis •e.cot) ° Then the group generated by 04 and contains an elation mapping 0.1. onto Ca. . coronan 1 Let m be an affine line of and let Trbe a collineation. group of Tsuchx that for every affine point

C E f2 there exists a. (C, ) -homol ogy. Then 63 is (mr\k ) 'e.00 )-transitive with respect to.

Kmx Corollary 2 Let 11 be a collineation group of Lr such that for every affine point P there exists a (P, -Q00)-homology Then e is a translation plane and At contains the translation group of e (see Andre Cip ) 25.

12 P is a finite projective plane and-Tris a collineation group of (? . If G" is (0,t.)-tran.sitive and (0,m)- transtive, e t mt then II is transitive on the lines of which do not contain. 0. (see Wagner L32J)

13 Let 11 be a finite projective plane of order n and let (i) possess a polarity 9. Then 9 has at least n+1 absolute points. If n is not a square then (1) 9 has exactly n+l absolute points. (ii) Ii n is even, then the absolute points are collinear.

(iii) if n is odd, then no three absolute points are

collinear. (see Baer E5] ).

14 Let U be a finite projective plane and let (I be any collineation group of P. Then the number of transitive classes into which -Ti- splits the points of G) is equal to . . • . • the number of transitive classes of lines. (see Dembowski E83 Hughes 5.,.1.3 , Parker 54] )

15 If (P is a desarguesian plane of even order n and if 6 is a conic of , then the n+1 tangents to e are concurrent at a point called the nucleus of 6 (see Segre ).

16 If U is a finite projective plane and if 0(.. is a collineation of , then the number of points of left 260 fixed by 0(.. is the same as the number of lines of Q left fixed by (see Baer D..] )

17 is a finite desarguesian plane of odd order. If 0( is a (A,a)-elation of &and if /3 is a (B,b)-elation of cf) (with A #-B, a41), A b, B a), then 4',1s> contains an involutory (a,Ab y.AB)-homology,. (see Mitchell )

18 Let 4kbe a collineation group of a finite projective space. Then the number of transitive classes of points under -TT is equal to the number of transitive classes of hyperplanes under

(see Dembowski [8],)

19 Let be a collineation group of a finite projective space Hk • If Tr-contains a non-trivial (1:y1k..1)-elation for every

hyperplane Hh_1 and for every PE.H1,t -1' then containsthe little projective group of H.

(see Wagner C3 ) 2?. 4 ON ELATIONS OF FINITE PROJECTIVE PLANES

To avoid unnecessary repetition in the statements and proofs of the first four lemmas, we shall use the term centre (axis) to

mean "centre (axis) of an elation of prime order p In the first three lemmas, all elations are assumed to have the same prime order.

Lemma 1 Let 6) be a finite projective plane and let-lrbe a collineation group of containing some elations of prime order P.

If no centre or axis is left fixed by, then7T-Is transitive on the centres (axes) of P,

Proof Let e.be any non-axis. For any centre PE an elation with centre P is an element of the group-FT,? which has prime ardor

p and which fixes P and no other point of e Hence, by Result 1, -Tr is transitive on the centres on e.

Lot m be any axis. If each centre QE.K1 has on axis distinct from m, then a repetition of the above argument shows tharic is

transitive on the centres of m, But, from Result 2, any axis

through Q is an axis for Q. Thus either-fr is transitive on the centres of3 or there exists a centre, R say, incident with only

one axis. Suppose that there exists such a point R, and let its axis be e,. Since the line joining R to any centre in (M, is a non-axis,-T17- maps R onto any centre of T\P...- . If R is the only centre of E, then the lemma if proved, so let P e be another centre. If Po( ' for. some.‹EIT, then we have already shown that -n---- contains an element, f3 say, such that Re = 0,0‹ and the

28, collineation/3cii maps R onto P. Hence eithenTis transitive on

the centres of CP , or there exists a centre, Q say, such that

QTrE (1 for all 11-01-. Since is not a fixed lino,- T1- contains:

an elation such that gzroer. If QeJE e then Q 2f. == Q i. e. Q is on the axis of But Q is not fixed by-n- and sin-I—must contain an elation, E say, moving Q. If Q62 e. then Qsdo..... Thus either Qeje e, or QS&O.and-11-is transitive on the centres of C? The dual of the above argument proves that-TT-is transitive on the axes of

Lemma 2 Let be a finite projective plane and let-rr-be a collineation group of containing some elations of prime order

P. If no centre or axis is left fixed by7TT- 1 thea--is transitive on the centre-axis flags of

Proof Sinc01-- is transitive on the centres of 63 each centre of G) has the same number of axes. Dually, each axis has the same number of centres. case (i) Each centre has only one axis

Lot (P, e) a,n.d (Q,m) be any two centre-axis flags. Fr om lemma 1, —77 contains a collineation OC. such that RD( = Q. But, by Result 7, & is an axis of Po( and hence, since each centre has only one axis, &C 7.17 m. case (ii) Each centre has more than one axis. Let t be any axis. For each centre Pc 11, contains an elation of order p fixing P and no other point of t So, by

Result 1, Tre, is transitive on the centres of E, Lot (P,m) and (Q,m/ ) be any two centre-axis flags. From lemma 1,1T contains an element e4 such that mo( = m and Po( E 110. 29.

Since m/ is transitive on the centres of m/ , there exists an

element f3eTcw, such that (Pct )p = Q. The collineation t

flags of .

Corollary Under the conditions of lemma 2, h, (where h e. is some constant), for all centres P and axes a. with PE e„.

Lemma 3 Let 6' be a finite projective plane and let bea

collineation group of q") containing some elations of prime order p. If 6) contains an axis with more than one centre and a centre with more than one axis, and if no centre or axis is left fixed by-TT- then the number of centres on an axis is equal to the number of axes through a centre.

Proof From the corollary to lemma 2, 1.;;)e., 1 = h for all centres

P and axes e. with PE But, since there is more than one 0 centre per axis , is a p-group (see Result 3). Hence h = p Pe, for some a.

Let t be the number of axes through a centre. and let tl be the number of centres on an axis. Let Pi be any axis and let P6., be a centre. Since P has more than one axis,q, contains an elation of prime order p fixing P and no other centre of e/ This elation must permute the other centres of e, in cycles, of length p. Hence t/ e 1 (mod p). Similarly t = 1 (mod p). Put t = 1 + kp and t/ = l+k p. We may assume , without any loss of generality, that , k/ k.

= group of all elations with centre P and

axis See last page of thesis. 30. Since ; =7 xy.e, 2_ e.„ and since = for P/Q, we have Pe(4,e_ Ige = (lc p 1) (PI -1) + 1„ But, from Result 3, 2--e, is a p-group

and so l el = p , for some b Thus (V p 1) (pa -1)+1 = Pb Let III; denote the subgroup of which fixes the line m. Then

1 orbit of m under feJe 1 2-e7 But where P = en me Hence I orbit of m under ; 1.1 2-p)el = te — If in is an axis and if P is a centre, then, since; must split the axes of P, other than C, , into orbits of equal length, we have

(orbit of m under ;1 = 4 where s?-1 •••••••• Combining and 0 gives 1,a+1 0 kp (pa - 1) + pa = pb %OWEN., 0 But we are assuming that ki>, k, and so a+1 kp (pa-1) + pa Thus kpa+1,4 skpa-1 skp + spa (since s?-1) and, hence, kp (pa +s) spa (kp+1). Thus since kp < kp+1 I pa+ s > spa i. e. S =

Substituting for s in gives k = pb-a-1 The dual of the above argument gives = (kp+1) (pa-1)+1 = pc and, since we are assuming k. k , we have c .c b. Substituting for k gives (pb-a+1) (pa-1) +1 - p b a b-a i.e• P 4•13- P = P

If b< 2a, dividing by pb-a gives pa + p2a-b-1 = pc-a-b which is, impossible since c + a> 2a> b. b-a 1 pb-2a,..7. pc-a If b> 2a, dividing by pa gives p which is impossible since c> a. a-1_ / Thus from b = 2a = ce Hence k = p k 9 and the 31. number of centres on an axis is equal to the number of axes through

a centre. Furthermore, this number is equal to pP4-1, where a e-1 - P We note here that if e contains a centre with more than one axis and an axis with more than one centre, then the condition that 17T- fixes no centre or axis is equivalent to the condition thatIT fixes no point or line. For future results we shall use the latter wording.

Definition Two elation centres P and Q are said to be connected if there exists a sequence of incident centres and axes connecting

P to Q "Connectedness" between two axes, or between an axis and a centre, is defined in the same way. The set of all centres and axes which are connected to a given centre P is called a connected set. (A connected set which consists of a flag only is described as trivial.) If a connected set contains more than one centre and more than one axis, then it is said to be a proper connected set.

It is easily seen that "connectedness" is an equivalence relationship. Consequently, if P and Q are any two connected centres, the set connected to P is identical to the set connected to Q, i.e. a connected set is uniquely determined by any one of its elements.

Lemma 4 Let be a finite projective plane and let-trbe a collineation group of containing some elations. If is any non-trivial connected set, then all in elations whose centres lie in.,2 have the same prime order. 32. Proof This in an immediate consequence of Results 2 and 3.

As a result of lemma 4, we can now associate a unique prim with any non-trivial connected set. This prime, which is the order of any elation whose centre lies in the set, is called the characteristic of the net. If 1= pa for all centres and ce axes belonging to a given connected set ig , then we say that ig has degree pa. In view of lemma 2, a degree is defined for a non-trivial connected set whenever no point or line is left fixed by TT. We observe here' that if no point or line is left fixed byTT

then every centre of a proper connected set has more than one axis

and dually i.e. the conditions of lemma 3 are satisfied. For the rest of this section we shall use the term centre (axis) to mean "centre (axis) of an elation ia-TI- ". Unless it is stated otherwise, we no longer assume that all elations have the name prime order. 33. Theorem 1 Let 9 be a finite projective plane and let-Tr be a collineation group of e containing some elations. If no point or line of es) is left fixed , and if Qi is an axis for more than one centre, then

(i) The group 2".pi_ e., is transitive on the axes, other than t through auy centre QE Q. Q A P. (ii) For any centre Rd-, is transitive on the centres, other than R, of the axes of R distinct from ti

Proof Since all the elations under consideration have their centres and axes in aconnected set, they must, by lemma 4, have the same prime order p. If each centre of has only one axis, then the theorem is trivially true. Suppose that each centre has more than one axis, then the conditions of lemmas 2 and 3 are satisfied. (1) Let 1;)e,1 = pa. Then, from lemma 3, the number of axes through any centre is pa+ I. Thus, if QEe, is a centre, the a number of axes of Q. which are distinct from e. is p . Hence, since no element of e, can fix one of these axes, is transitive on the axes of Q other than . (ii) From lemma 3, the number of centres on any axis of Q is pa+1. Thus is transitive on the centres, other than Q, of any axis m through Q, m e.. But, from part (1), ;e, is transitive on the axes of Q other than e., and the theorem is proved.

Corollary Under the conditions of Theorem I, 11 is doubly transitive on the centres of an axis , 34. Proof For any centre PE Q.. p is transitive on the remaining centres of ei .

Theorem 2 Let 6) be a finite projective plane and letTr be a collineation group of 6) containing some elations. If no point

or line is left fixed by -11- and if (9 contains a proper connected

set , then, fc,r any line es,s , the point orbits of G),.e. under 2e are affine subplanes with special line and with the centres

of as special points.

a Proof Let the degree of .9 be p . Then, by lemma 3, the = p2a for number of centres on each axis of .2 is pa+l and

any axis e.E..8 LetT be any point ofBe". Then the- length of the orbit of is p2a. T under Let X and Y be any two points of (94,which

lie in the same orbit under , and let XY4t= R. The elation which maps X onto Y fixes the line XY and must, therefore, have

centre R. Thus the line joining any two points of the same orbit must intersect e.„ in a centre. To prove this theorem, we shall show that the set of points of n an orbit under ; together with the centres of " form a projective subplane of 0 the lines of which are the lines of 6

To do this we must show that

(a) there is a unique line joining any two points of the set

(b) any two lines of the set intersect in a unique point of the set (where a line of the set is defined as any line joining

two points of the set)

(c) the set contains a non-degenerate quadrangle. Since the points of any orbit are points of e , (a) is obviously 35 true. .Also since a- contains at least three centres, (c)

is satisfied.

Let A, B, C, D be any four points of 61Lwhich lie in the same

orbit under and which form a non-degenerate quadrangle. Let ABA CD = E, let ABA P = P and ODA= Q. We have to show that E also lies in this orbit. Suppose it does not. Since the length

of the orbit of A under is pa, the line AB contains pa points

of P\ vrhich belong to this orbit. The line joining C to any one of these points intersects e, at a centre. But, since we are assuming that E is not a point of the orbit, Q is not one of these centres nor, obviously, is P. Thus E has pa+ 2 centres. This contrdiction shows that E must be a point of the orbit and completes

the proof, of Theorem 2.

Theorem 3 Let be a finite projective plane and let t T be a collineation group of 9 containing some elations. If contains an axis e„ with more than one centre and a centre PE twith more than one axis, and if P is not fixed by then the centres of

the axes of P are the points of a desarguesian, subplane of (9

Proof Since P is not fixed byTr , Tr cannot fix any point or line. Thus, from Theorem 1, is transitive on the centres / other than P, of the axes of P distinct from Hence, from Theorem 2, the centres on the axes of P form a projective subplane, say, of Q. But, from Theorem 1, Q is (Q,QP)-transitive for every point QE , Q•P, mid so, by Result 5, lt" is desarguesian. Thus we can now associate a unique desarguesian subplane with any centre P which satisfies the conditions of Theorem 3. We denote this subplane by 2 13 . Although each point ofP is a centre

36 which is connected to P, is, not necessarily the complete set connected to P. Our next lemma provides a condition under which every centre which is connected to P will lie in I.133

Definition The centres P, Q, R are said to form an elation triangle if (1) they are not collinear, and (ii) the lines

PQ, QR, RS are all. axes.

Lemma 5 Let 63 be a finite projective plane and let-TT be a collineation group containing some olations. If a connected set contains an elation triangle, then that connected set is a desarguesian subplane 13)/ and Tr restricted to IT contains the little projective group.of V.

Proof Let P, Q, R be three centres forming an elation triangle. Since R has more than one axis and PQ has more than one centre, cannot fix a point or line of . So, from Theorem 3, the centres of the axes of P form a desarguesian plane 112 . Let €-= PQ and mr; PR. The group 3t. is transitive on the. 7 centres of m other than P. Thus the line joining Q to any centre

of m is an axis i. e. every lino of .Pi• which passes through Q is

an axis. But, from lemma 3, the number of centres of m is equal to the number of axes of Q, and so every axis of Q is a line of From the proof of Theorem 371; is transitive on the points of (Pi \ P, and so every line of [.P} is an axis and every axis. of any point SE. 1)} is a line of Thus 1P is a complete connected set. Consider -11-restricted to For any flag (Q,m) in.{A contains a (Q,m)-elation. Hence, by Result 6 Trcontains the

v. little pr ojectivo group of (P1

Lemma 6 Let e be a projective plane (not necessarily finite).

If oc is an (11,a)-elation and if k is a (B,b)-elation such that A # B, a b, then the only point of 9\ AB left fixed by die is a^ b.

Proof Let P be any point of \ AB, P e...A b, Then P, Po( , I. and

Pd 2 Pegg , B are two sets of collinear points. If Pociez:P, then fiRcf3 A, B are collinear, i. e. P E A B. This contradiction shows that

00 cannot fix P and proves the lemma.

Lemma 7 Let e be a finite projective plane and let-II-be a collineation group of If containsan (A,a)-elation 04 and a (B,b)-elation /3 with 14B, a b, then ) if 1 A,c,1=1;_jb k> > contains two elations d such that A2S'i 4=-• A.

Proof Let ( I denote the group <;,(1.3;,,f), let e.1,33 a/Ida:x(1M). Then every elation in I 1 has its centre one. and its axis passing through 0. Let k be the number of centres of elations inTT . The number

of images of A, distinct from A, under ZB b is h-1. This is true for all centres PE e , P#A. If A is not fixed by the product of two elations in , then these images must all be distinct and there must be at least (k-1)(h-1)1-1- centres on e. Thus (k-1)(h-1)+ 1 4 k or (k-1)(h-1)4 k-1. Since k is greater than two, this implies (h-1) 1 which contradicts the assumptions of the lemma. Hence -r-r/ there exists two elations in 1. such that A'di z A4. i.e.

such that A b't A. 38.

Lemma 8 Let be a finite projective plane and let k.i/ be a subplane of 6) . If fX is. a (P, e,)-elation of 6) such that (1) P e E xm/ (ii) there exists a point L £6 , E such that LAI LT-, then in/ 0( ie a (P, e..)-elation of (i'd

Proof Let Q E. ebe another point of and let m be any line of

through P such that .t..V m. Let B m Then B E Q and

Bat 6):

The points ^ and B, together with the points of e. in k3" f / generate 0/. But tY is also generated by l,.aC , Bo( and the 4.1/ points of e, in (r. Hence Ch. leaves (...11 invariant. Consider 'X cr„, as a collineation of Since c 1k. , 0f is not the identity on en/ But QC fixes e. pointwise in kJ' and fixes P linewise in tI Thus Ot... is a. (p,e. )-elation of

Lemma 9 Let CP be a finite desarguesian plane of order pk and let-Trbe a collineation group of (3) fixing a flog (P,t). If ire fixes more than two points of 2 then the number of points of left fixed by TT is pi" + 1 where id k.

Proof Suppose that Tr fixes the points P,Q,R of Since k is desarguesion., we may coordinatize it with elements of GF(p ).

Choose a to be the line xz o and let Pr. (0,0,1), Q= (0,1,0 ), R (0,1,1).

If fl- is a linear transformation, then Tr must fix every point of

e. and the lemma is proved. Suppose cro( where o( is a linear k transformation and cr is an automorphism of GF(p ). Then

(x l y, ()ca.. y`r ) tx.. Hence, since for any automorphism 39.

of a field,06= 0 and 16:= 1, PTrz: Pokz.-P, QTr = Qoc= Q, RTT lick= R and c< must fix every point of a Thus, for points of e.,Tr.-za- and (o,y,z)jr:-.- (o,y`r , za- ). Since 63 is projective, (o,icy,kz) (o,y,z) for k*o and so the points of e\ P are of the form (0,1,z). Thus, since (0,1, z )1r (0,1, z`r ) , the number of points of e P k which are fixed by ir is equal to the number of elements of GF(p ) fixed by o- . These elements form a subfield and, hence, there are / k p of them, where k. This proves lemma 9.

Theorem 4 Let be a finite projective plane and let Trbe a collineation group of ER containing some elations. If no point or line is left fixed byTr and if e contains a proper connected set 26 of degree pa, where pa> 2, then is a desarguessien subplane such that (1) the centres and axes of all elations of order p are contained in and. (ii)restricted tot contains the little projective group of 2

Proof Throughout. this proof we use the term elation to mean

"elation of order p". Conequently, the term centre (axis) will mean "centre (axis) of an elation of order p". The theorem says, essentially, that the join of any two centres is an axis and that the meet of any two axes is a centre. Lot P be a centre of S with axis in, and let Q be any centre which does not lie on m. Vie suppose that the line e.= PQ is a non-axis and shall show that this leads to a contradiction. From lemma, 3, the number of axes of Q is equal to the number of

centres of m. Hence, since PQ is a non-axis, Q must, have an axis,

111 / say, such that in m is a non-centre.

40. Let 0 = mAm i and let -rr= Since no P) PC)) c./q() > * point or line is left fixed, Tr is transitive on the centre-axis

flags and i ivt,1= h for all centres A end axes a with AE a. Consequently, since it= pa, (where pal 2) , for centres of ,2

1 ,i.)0.1.:..p- a for all centres and axes of . Thus, from lemma 7,1-T contains two elations, of andp say, such that Pocp = P. By Theorem 3, the centres on the axes of P are the points, of a

dosarguosian subplane of order pa which we denote by Pj . Since {PJ and 0 iPj, the only elements of fp3 left fixed by tile

are P and in (see lemma 6). Let 2= diS and let-17 •

We now consider1-1 as a collineation group of 1:)..3 . Suppose that „e d fixes a centre AE m, A*P, then, since y fixes no centre of m

other than P, must also fix a third centre, B say, of m. Hence, by lemma 9, the number of centres of in which are fixed by 0 is

p 1 where cia. If t is the smallest power of 'e which fixes A,

then must divide these fixed centres, other than P, into orbits of equal length pa , where c1 is some constant satisfying c1 c. c Hence t = p . The number of non-centres of m which are moved by e is n-pa-1.

If 3 fixes any orbit of non-centres on m, the length of that orbit must divide t i.e. it must be a power of p. But, since pin,

pin-pa-1 and so 0 cannot fix all the non centres of m. Thus Kt*I- on P. case (i) c= a (i.e. 2f t fixes every centre of in)

ince e 10 a collineation of fixing m pointwise, either (a) 2Sb is a homology of iP3 or (b) ?fe is an elation of {133 or (0) 2fr is the identity on [P1 41 Suppose (a). If efixes a point, X say, of {P3 '\ m, then 1S t fixes the line PX E 11'3. But r is a permutation group on the lines of iP} through P, other than in, such that 2s leaves none of them t fixed. Thus e must fix a third line of 13../ through P and must, therefore, be the identity on. fri .

Suppose (b) Thrs ( )P=.- I on [P3 but is not the identity on@

since ?f tp# I on m. Thus we may suppose that of is the identity on . The configuration fixed by et is a subplane, e say, which contains fP} . Since 0 E and 0 P , (:S4 FP3 . Let the order of LT be h. Then the number of points of m which are left fixed by e t is h+ 1. But, since t is a power of p and since ?I' fixes two points of m, h+ls 2 (mod p) i.e. 11E1 (mod p). If 5. is a (P,m)-elation of si9 , then E fixes all the points of in which belong to kJ' . Hence, sinceZpmis transitive on. the centres of an axis of P and since all the centres of the axes of P

are points of 6 is a (P,m)-elation of (.5 (see lemma 8).

But c has order p and pi h. This contradiction shows that our assumption that Ci was a non-axis was false. case (iij c < a

Let 1/,---.2C- t. Since fixes at least pc+ 1 points of 1,123 , must fix at least pc-i- 1 lines of [1:' (see Result 16).

If fixes a point X {P3 , such that Xt ra, then, as in proof of case (i), must fix more than two lines of 1P1 through P and must fix a subplane of order a power of p.

Suppose does not fix any point of {P1 • in. Then fixes p÷1 lines of IP} through some centre, A. say, of in. Let m/ be one such line and consider <21.> as, a collineation group of fixing

42 the flag (A,m ). As in case (1) there exists an integer 3 such

that fixes pd-t-1 points of .{P3 which belong to re, (where a is

a power of p and d is some integer satisfying d.6. a.). Since , where st is a power of p , aS is not the identity on 9.

Hence, as in case (1), fixes a subplane, of , . Let the order of Li" be al. Then, as before, nit- 1 (mod p).

Let X, Y be any two points of 01/ which belong to w such that

XY, X m, Yi in and Xii=YA. Since, by Theorem 1, 2m is transitive on the points o4Pj N m, there exists an (A,m)-elation I/ of e mapping X onto Y. By lemma 8, this is also an elation ofe which is impossible since p''zi Thus the line PQ must be an axis.

The dual of the above argument shows that the intersection of

any two axes is a centre.

Thus all the centres and axes are connected and must lie in /8 Hence, since 2 must contain an elation triangle, lemma 5 proves the theorem.

Definition Let be a permutation group cn a set S. If the only element of which fixes any element of S is the identity, then we say that is one-regular on S. We note that need not be transitive on S.

Lemma 10 Let Q be a finite projective plane. If oZ is an involutory (A,a)-elation and if le is an. involutory (B,b)-elation such that AltB, a b, then <0(p> is, one-regular on the points of

(S) which are not on AB, excluding an b.

2s Pr oof Since ( oq:3 ) -7: pd,.... ( . . (3) , where each bracket contains 2s-1 terms, and since (dp )28417 0( (00(.. . . ita

43 (0/43..... 0), where each bracket contains 2s terms, we have that, t for any t, ( ate ) is equal to o< times a conjugate of either 0( or

ea that (cxp )t is equal to the product of two °lotions whose

centres lie on AB and whose axes pass through an b.

Let the length of the smallest orbit of points of 6)\.A$,

(excluding 0.A b) under <=q3> be r and lot R be a point of this

orbit. Then (o(3 )r fixes R and so, by lemma 6, (die I. Thus

all orbits have length r and <0(13> is one-regular on the points of 6), LB, excluding an la.

Theorem 5 Let q) be a finite projective plane of oven order, and

let-1T be a collineation group of containing some involutory

elations. If no point or line is left fixed byTEand if e?

contains a proper connected set ig of degree 2, then either

(i) is a subplane of order two andif restricted to contains the little projective group of

or (ii) is a subplane of order four minus an oval and its dual andli restricted to 2 is isomorphic to either A6 or S

Proof If contains an elation triangle, then lemma 5 proves case (1) of this theorem. We shall henceforth assume that 2doez

not contain an elation triangle.

Since this proof is rather long, we split it up into a sequence

of remarks.

Remark 1 If is any axis in ,g with centres X, Y, Z, then the

axes of X, Y, Z determine four unique non-centres of

These four points, together with X, Y, Z, are the points of a.

subplane of order two, 44

Proof Let x1, x2 be the other axes of X, let yl, y2 be the other axes of Y and lot z 1 be the other axes of Z. Let P = xin Q =xl1 y2, R = x2A , Sr; x2^ y2. Since we are assuming that there is no elation triangle, P,Q,R,S are all non-centres.

An (X,x1)-elation fixes P and Q and maps Y onto Z. Since YP and

YQ are both axes, the lines ZP and 2Q must be axes of Z.

Similarly ZR and ZS are axes of S. We may assume, without any loss

of generality, that zi= PS and z2 QR.

Consider the configuration formed by P,Q,R,S,X,Y,Z and the axes

of X,Y,Z. Any two of these points are joined by one of those

axes, any two of the axes intersect in one of these points, and

P,S,X,Y form a non-degenerate quadrangle. Hence the configuration

is a subplane of order two. Remark 2 If E is any axis of 2 and X is a centre of e then X determines two unique non-centres on

Proof Let X1, X2 be the centres of x1 other than X, and let X3, X 4 be the other centres of x2. An (X3, x2 )` . elationfixes R,S. Since

it maps x1 onto , it must map X1, X2c. onto Y,Z . Hence, since YR, YS, 2R, 2S are all axes, the axes of Xi and X2 arc X1R,

X2R, X2S. Similarly the axes of X3 and X4 are X3P, X3Q, X4P,

X Q. 4 Let XIRAe.--- M and X2Rnk= N. An (K, e. ) -elation fixes 111,N and interchanges X.1 and X2. Thus XiN and X2M are also axes. A (Y, e, ) -elation fixes 11,N but maps x1 onto x2. It therefore maps

{Xi X2i onto (X31 X. Thus the axes of X3, K4 are X3la I X3 N, X4111 X4N and every axis of Xi , X2, X3 or X4 intersects 4- in either 14,N or X

z21 = the set consisting of Xi and X2 45

Since e, has only three Lcentres and since we are assuming that /2

does not contain an elation triangle, N and N are non-centres. Remark 3 Let e be any axis with centres X, Y, Z and let the two unique non-centres which are determined by X be 11 and N. If ono of the non-centres of , determined by Y is N, then the other is N.

Proof Let the two unique non-centres which are determined by Y be RI and N . A (Z,z2)-elation interchanges X and Y and, therefore

maps 11,N onto 11,141 * Since 11 is not fixed by this elation, N

must be mapped onto N', and N must be mapped onto N. Thus , since

the elation is involutory, N= NI.

Remark 4 (i) For any non-centre 0 on mare than one axis, and for any axis Q. which contains 0, there exists an involutory (A,A0)-

elation c< and an involutory (B,BO)-elation 3 , with A0(1 , such that

(ii) Since o( f3 fixes the line AB, the point P. tA AB is. 3 also fixed by dgo. Then P is a non-centre and ( dy3 ) = I.

Proof .(i) Lot the number of axes through 0 be k. Since t;1= 4 for any axis, the length of the orbit of e. under is four for any axis m, (m*1). Thus EL has 3 (k-1) -4r1 images under el ati ens with axes passing through. O. But 3(k-1)+1> k and, hence, there exist elationa oils and (3 such that ea P(a i. e. 43=P. (ii) Suppose that P is a centre. Then OP has two centres, Q and a R say, which are not fixed by otp „ Thus (0(0) fixes Q and so, by 2 lemma 10, (cV) =I. But <.°(3>is4 a permutation group on the n-1 points of e, other than 0 and P, such that 0(13 fixes none of those points. This is impossible since n-i is odd. Hence P is a non-centre. The group <00> is a permutation group on the three centres of 46

and so, by lemma 10, (o93)3= I.

Remark 5 If Q, is any axis with centres X,Y,Z then tho two unique non-centres of e. determined by X, as in Remark 2, are identical with those determined by Y and with those determined by Z.

Proof Let h and N be the two unique non-centres of P., which are determined by X. An (X,x2)-olation interchanges LT and N and also interchanges Y and Z. Thus, if M and N are also the two unique centres of e- whichare determined by Y, then Z also determines M

and N. Thus, as a result of Remark 3, the number of non-centres of e. determined by X,Y,Z as in Remark 2 is either two or six. If it is two, then the remark is proved. Suppose that X,Y,Z determine six non-centres on and let 0 be

one of these points. By Remark 4,T contains involutory elations

of and , whose axes pass through 0, such that eo(prl and Ke= I. 3 Since dip permutes the centres of Lmnongst themselves, it must also

be a permutation of the six non-centres. Thus, since c(6 fixes 0,

at least three of them must be fixed byegoi,„ But this is impossible

by lemma 6 and the remark is proved. Remark 6 Associated with any axis e are six unique non-centres. Four are determined as in Remark 1, and two as in Remark 5.

Remark 7 Associated with any axis, is a unique subplane of

order four. Proof Let X,Y,Z be the centres of e. and adopt the lettering of Remark 2 for the axes of X,Y,Z and their centres. Let P,Q,R,S be

the four non-centres of Remark 1 and let M,N be the non-centres of

Remark 5. We now have a set of twenty one points. From the proof of Remark 2, the line joining any two of the six 47 non-centres is an axis and contains either. X or one of the X.

(i.=1 ,..J4). Hence since these six points are independent. of X,Y,Z, if the line joining any two non centres is an axis for one of the X. but does not contain X, then this line is also anaxisforoneoftheY.and one of the Z. (L --- 1 ...4) Thus the line jvining any two non centres is an axis which contains, five points of the set. Furthermore, any axis. of a centre of the set contains two of the non-centres.

Let m be a non-axis joining Xt.,(1:.=1 or 2) to Xe (j=3 or 4).

Since; contains an elation mapping X i: on to X., mAt is a centre.

We suppose that mnQ= Z. A (Z,z2) elation interchanges X and Y

but fixes m. Thus zany]. and m6y2 are centres and, hence, m contains five points of the set. For any fixed j (j=1...4) there are only three lines through

X.whiclic.ontainoneatheY.(i:r1...4). Two of these are axes

andthethird,whichcontainstwooftheY.is the non-axis joining X. to Z. Each of these lines contains five points of the set. Thus our set of twenty one points is such that the line joining any two of the points contains exactly five of them i. e. they are

the points of a subplane of order four. Remark 8 The centres connected toe, are the points of a subplane of order four, minus an oval. This plane is desarguesian and the oval consists of a conic and its nucleus. Proof Let ebe the subplane of Remark 7. The axis of every 4N/ centre of ur is a line of T and its other two centres are points of Hence, no centre (axis) of T can have an axis (centre) 611 not in tr , and the set connected to is contained inlY • Since no three of the non-centres are collinear, Result 15 proves the 48 Remark. -rir/ Remark 9 If fi is-Trestricted to then 11 A6 or S . e, 6 Proof Since the conditions of lemma 1 are satisfied, Tris 4-Y transitive on the centres (axes) of W. Since the number of i 't7J 1 —rrt centres oft...Il ia 15, ITO 15.111c1 1 where 114: is the subgroup of 11 fixing a centre. C. By Theorem 1, is transitive on the axes through C and on the non-axes through C which lie in LT . Thus 15,3,2 I I j-t. I where ( LQ , is the subgroup of fixing an -r-ri axis through C and a non-axis (. through C. But E.,Q,E contains three involutory elations which, together with the identity, form 1,/ a four group. Hencellp2 1= 4h and ill I = 360 h (where h is some constant).

Suppose that there exists a collineation Tic:II such that IT fixes, the oval of non-centres pointwise. Since four points of the oval form a non-degenerate quadrangle, IT must fix a subplane Qof of order two. This subplane cannot contain any other point of the oval since it does not contain five points such that no three are collinear. Hence, since a subplane of order two is maximal in lYQ

TT must fix the whole plane and I must be faithful on the oval.

But the only groups of order 360h which are faithful on six letters are A6, when the order is 360, and S6, when the order is 720. This final remark completes the proof of Theorem 5. The fact that this. situation actually does occur is shown_ in ?6J. (This paper is also submitted as part of the thesis).

Theorem 6 LetEP be a finite projective plane of even order and let IT be a collineation group containing some elations. If a proper connected set d2 of degree two is a subplane of order four minus an oval and its dual, then the centre and axis of every 49 involutory elation is contained in

Proof Throughout this proof, all elatione have order two.

SincelTcannot fix a point or line, 12- I 2 for every centre P f:21e. = and every axis e with PE..e.. Let e be an axis with centres X,Y,Z and with associated non.- centres 14,N (determined as in Remark 2 of Theorem 5). Let 0 be any point of e such that 0 is on more than one axis. Prom Remark 4 of Theorem 5, Tr contains. an (A,AO)-elation a. and a (B,BO)-elation p , A Q, B g such. that 6(8 r•E and 60--• I. Since 40 is a permutation group on the centres of e., , cies must permute M and N amongst themselves i. a. cif.3 must fix M and N. By lemma 6, this is

only possible if 0=M and N= AB A t. Suppose there is an axis m which does not belong to ,,g and let Let .ke m be a centre and let the other axes of A be a1 and a2. Since we are assuming that mi.2, a_ing. and a2A are non- centr es. But we have already shown that there are only two non- centres of 2. with. more than one axis through them. Thus m and the Theorem is proved.

Lemma 12 Let be a finite projective plane and lot Q be a sub- plane of (3) If 0( is a (P,e..)-perspectivity of lT such that 0C. leaves v invariant, then P€ (3 Qe. Q andok. is. a (P l e.)-perspectivity in/ of U'.

Proof Let m be any line oft..TQ Then mad is also a line of and, thus, memo( is a point of E? which belongs to e Let m/ be ra/ any other line of such that MA MK . Then m nctu4E and

MiA floc e T. Hence e E (9/

50 Similarly, by the dual argument P

The proof of lemma 8 now proves this lemma.

Theorem 7 Let Pbe a finite projective plane and letTT-be a collineation group ofP containing somo elations. If no point

or line is left fixed by and if Rontains a proper connected

sot 2 such that ether (i) the degree of is greater than two, or (ii) ,2 is a subplane of order four minus an oval and its dual, then all elations of Q have the same prime order.

Proof (1) is an immediate consequence of Theorem 4 and lemma 12.

(ii) is an immediate consequence of Theorem 6 and lemma 12.

Theorems 4, 5, 6, 7 form the major results of this section. We now use these results to prove some less general theorems. We begin by considering planes which are semi-translation planes

but are not strict semi-translation planes. 0stromD1 has proved the following: "He, a strict semi-translation plane of order q2 (where q is a prime power greater than 2) with respect to and nr if e admits a collineation moving P.., than e has a subplane of (71/ order q such that (i) Every collineation of & carries 'T into

itself. If is a strict semi-translation plane I / with respect to any line t , then t belongs m/ 4 / Am/ to and the centres of e are points of LY (iii) Every point of G) is a centre for some line

of (3- ." As a result of Theorem 4, we can now determine what happens if the "strictness" condition is removed. In order to avoid tedious

51 repetition, we shall assume that our planes are definitely not strict.

Lemma 13 Lot be a finite projective plane of order n. If h>1 for all points Pee, P # 0, then 14,,J7----n.

Proof From Result 10, ;contains a non-trivial (0, PA-elation. Let ed=k> 1. Then (cj n(h-1)+ k. Now ('-e I= (;,,,e j.lorbit of m under;_ where PE_m, m Hence j;_( = hs. Equating the two expressions for I; ) gives n(h-1)-+ k =hs or nh-(n-k)= hs. Thus, since sin, nh-(n-k)han, and so, either nh-(n-k) -“nh or n Suppose ii k. Then • knh‘. n-k or k n-inh. But h 2 by assumption, so k 40. This contradiction proves k =no

2 Theorem 8 Let pbe a finite projective plane of order q , where q is a prime power greater than two. If is a semi translation plane with respect to and if Q admits a collineation moving then either

(i) is the dual of a translation plane or is desarguesian.

Proof Let-rbe a collineation group ofq)which is generated by the elation with ,Aq X i S P... and a collineation which moves 'e- A stronger form of Result 10 (see Theorem 1, Ostrom a33) states that if .;1>. n,, then. every, point of tis an elation centre. Thus, since Q is not a strict semi translation plane with respect to every point of ., is the centre of an elation in 1r

where scq 52 case (1_,) Tr fixes a point 0.

Since 0 is fixed every line of 9 \ 0 is a non-axis. Hence, since5from the- proof of lemma 11-7 is transitive on the centres of a non-axis, Tris transitive on the centres of ci3\ O. Thus, 1k:h >1 for all centres PEe•C). In particular 1...p.e.17---h for all. Pe € PILO and, since e_ is not gized.7 1; ri‘ l=h 1.r all REm, R 0. Thus, by lemma 13, lT is (0, )-transitive and (O,m)-transitive i. e. is the dual of a translation plane with respect to 0. case (ii) T -does not fix a point or line. The conditions of Theorem 4 are now satisfied, and, since every point of is a centre, is desarguesian.

Theorem 9 Letq)be a finite projective plane and let bea collineation group of 6) such that every point of is the centre of an elation init • Then either - (i)lffixes a. point of 6:3 . Then GI is the dual of a translation plane an.d7- contains the dual. of the traniaation group of 9. or (ii)ldoes not fix a point of e • Then e is desarguesian andTrcontains the little projective group of 6).

Proof Let 0( be a (P,. )-elation and let le be a (Q,m)-elation. Since eA m is a centre, the flags (P, e_) and (Q,m) lie in the same connected set, and so, by lemma. 4, op( and le have the same prime order. Hence, all elations inlThave the same prime order. case l fixes a point 0. The proof of case (1) of Theorem 8 proves this case. case (La II does not fiX a point of 53 In this case the conditions of Theorem 4 and 5 are satisfied.

Hence, since every point of G3 lies in the same connected set, (9

is desarguesian and --icontains the little projective, group of (9.

Theorem 10 Let be a finite projective plane of order n and

letTr be a collineation group of U If for every flag of there exists an e' ation in fixing that flag, then either

(1) (1 fixes a point of G) - Then (32) is the dual of a translation plane and Trcontains: the dual of the translation group of p. (ii) Ti fixes a line of e • Thene is a translation plane and- TT contains the translation group of (5). (iii) does not fix a point or a line of e • Then is desarguesian. If n 4, thenTr contains the little projective group of , while if n=4 11- either contains

the little projective group or is isomorphic to A6 or S6.

Proof We note that if the flag (P,, ) is fixed by an. elation, then

either P is a centre, or g is an axis. Hence every point on a non-axis is a centre and every line through a non-centre is an axis case ji) ir fixes a point 0. Every point Pi..e\O is the centre of a. (P,OP)-elation. Hence, by Result 10, 0 is a centre and the conditions of case (1), Theorem 9 are satisfied. case (ii) Ti fixes a line Q The dual of the above argument proves this case. case (iiil -1-7- does not fix a point or line. We must first show that all elations of 1 ? have the same prime 54 .

order. We prove this by contradiction.

Let cA. be a (P, el-elation of order p and let A be a (Q,m)- elation of order q where q p. Then Q,\m is a non-centre and. AB is a non-axis. Since p #q, at least one of p or q is not equal

to 2. Suppose p# 2. Since Qn m is not a fixed point, @contains another non-centre, 0 say, and, since AB is not fixed, 6) contains

another non-axis, in/ say.

Since P and P.- are not fixed we may assume Pc/ m/ and. 0 1 'E.

Every line through 0 is an axis and every point of m1 is an axis,

thus PO is an axis of P and m/AQ, is a centre for e. . Hence P

belongs to a proper connected set of degree greater than two and

hence, by Theorem 7, all elations of have the same prime order.

Since—n- fixes no point or line and since, by the above argument, e contains a centre with more than one axis and an axis with more than. one centre, a connected set is either a subplane or a subplane

of order four minus an oval and its dual.

Let kt" be a connected set and let Q be any non-centre not ni belonging to V Since Q is a non-centre, every line through

is an axis. Let P and P be any two points of W such that QP #•QPia Then, since QP and QP' are both axes, QP and QP, 4-1 are both lines of (3' and, hence, Q E Ti Thus (9 and Theorems 4 and 5 complete the proof of Theorem 10. 55 PERSPECTIVITIES OF FINITE PROJECTIVE PLANES..

Lemma 14 Let p be a finite projective plane and let Trbe a collineation group of e such that every point of (Y) is the centre of a homology in -T E. If a line, e,, is the axis of homologies for more than one centre but no line possesses three non.-collinear

centres, then (5) is (eA ni,e) -transitive and ( e A m,m)-transitive (where m is the line joining the centres of e., ),

Proof Let Cl and C2 be any two homology centres with axis e.„ Let D be any point of e other than eAm and let > be a homology with centre D. Since ' C1A and 02 X must belong to m i.e. m must be the axis of A . Thus every point Pie , Pi m, is the centre of a homology with axis m and hence, by the dual to Corollary 1 of Result 11, e,m,m)-transitive. By interchanging the roles of eand m, a similar argument shows (S) is RAmj., ) -transitive.

Corollary Under the conditions of the lemma, Tris transitive on the lines of Y) which do not pass through m. Hence, if no point or line of is left fixed, every line of (f) is an axis and, by Result 8, is desarguesian.

Lemma 15 Let be a finite projective plane and let1T—be a collineation group such that every point of is the centre of a homology in-r1 T . If a line, e is the axis of homologies for three non-collinear centres, then-TT- contains a (1,,,e, ) -elation for every point PS e_

be three non-collinear homology centres of Proof Let Cl, C23 C3 56

Choose any point PE e. and let A be a homology with centre P.

Since cannot fix C1, C and C we may assume C 2 3 l)4 Old Then C and C are both centres of homologies with axis! 1 IA and, from Result I IT contains. a (Ci.CiXA , But Cry AR_ =P.

Hencelicontains a (P, e)-elation for every point P6 e.

Theorem 11 Let be a finite projective plane and let if be a

collineation group of If for every point PCQ there exists a homology iniT with centre P, then either (i)-Frfixes a line of e Then (P is a translation plane and Trcontains the translation group of T. (ii) Tr fixes a point of CP . Then the conclusions are the dual of those for case (1). -TT- fixes no point or line of 9 Then lT is desarguesian and f' contains the little projective group of V.

Proof case (i) ir fixes a line, Q.

Every affine point is the centre of a homology with axis etc)

Hence, from corollary 2 of Result 11, 1.3 a translation plane and 11 contains the translation group of e. case (ii) 1fixes a line, 0. We first note that the axis of every point, except of course 0, passes through 0. We now consider the following three possibilities

(a) An axis P. , through 0, has only one centre, P. Let D be .any point of e, other than 0. Then D is the centre of a homology A whose axis passes through 0. Since ex.- a.) P must be

* This case has been proved independently by Miss Coffman. \ 57 fixed by A and the axis, of A must be OP., So every point DEP__ D*0, is the centre of a (DIOP) -homology and hence, from corollary: 1 of Result 11, 9 is (090P)-transitive. But OP is not a fixed

line, so is also (0,0Q)-transitive where QOP. This means, by Result 12, that- rr-i. transitive on the lines of QP which do not pass through O. Since the axis of 0 is one such line, every line of which does 1Lot pass through 0 is the axis of a homology with centre 0 and so, from the dual of corollary 2 of Result 11, Q? is the dual of a translation plane and fl contains the dual of the

translation group of 6). (b) An axis P, , through 0, has mare than one centre but has no three non-collinear centres. Let m be the line of centres of e, and let D be any point of C, other than 0 or eA m. A homology with centre D fixes R, and so as in (a), must have axis m. But the axis of D passes through 0.

Thus OE m and, as in (a)„ Q is (0,m)-transitive. But m is not fixed and so, as before, Q is the dual of a translation plane and -Fr-contains the dual of the translation group of

(c) Every axis through 0 has three non-collinear centres. It follows immediately from lemma 15 that for every point PE C9

PO O, -rrcontaina a (P,OP) -elation. Hence, from Result 10, every point of G.) is the centre of an elation in ITand the conditions of Theorem 7 are satisfied.

case (iiil No point or line of (9 is left fixed by 11 In view of Result 8, we assume that some line of e is a non-axis. Consequently, some axis of g) has more than one centre. There are two possibilities: - (a) No axis has three non-collinear centres. 58 From the corollary to lemma, 14, e is desarguesian and contains the little projective group of G) (b) Some axis, e., , has three non-collinear centres. From lemma 15, -Trcontaina (P, e_ ) -elations for all P E. Since e is not fixed, —1T also contains (Q,m)-elations for all QE m, where m-fC,. The centres of these elations are connected and hence,

by lemma 4, have the same prime order, p say. But no point or line

of C is fixed byir so, by lemma 2, 1;c1= pa >1 far all P E P._ and pa) 1 for all QE m. Thus, from Result 4, is a trans- lation plane with respect to 2. and m. Consequently since e A is not fixed by7r, (9 is desarguesian and contains the little projective group of (P,

Lemma 16 Let (5) be a finite projective plane. and letTh be a collineation group of (3). If PE Q. is the centre of a (P,m)- homology in-TT and ifTr contains a (C,,e, el ati on for some point Cm, OP, then (i ) contains a (P, e. )-elation (ii) TT contains.,ia, RA in, -e.)-elation.

Proof Let c/‘ be a (C,e_ )-elation. Then Po( = P, but mo( m.

So, from the dual of Result 11,11- contains a. (P,P.m Arno( )-elation, i. e.1 contains a (P, )-el ati on. Suppose71 — does not contain a ( '\m, € )-elation. Let ./aj...k and let k. Then k > h. Since Q A m is a non- centre, m has k distinct images under .2-e_ and so P has k distinct axes passing through mAQ... But, from the dual of Result 11, for any two axes m and m/ T contains a (P, e. )-elation mapping m onto 59

Thus h k. This contradiction shows that contain a

(e A m, Q )-elation.

Theorem 12 Let (c) be a finite projective plane of order n and let --rr be a collineation group of (P such that every point P is the centre of a perspectivity inTr and every line .eE6--) is the axis of a perspectivity in . Then either (i) Tr fixes a line of Q . Then @ is a translation plane andl-r contains the translation group of E. (ii) TT fixes a point of o Then the conclusions are the dual of (1)

(iii) 11— fixes no point or line of V Then, if n is not a square, V is desarguesian andTi contains the little projective group of e

Proof case (1) (1 fixes a line, , Every affine point must be the centre of a homology with axis (0

Hence, from corollary 2 of Result 11, is a translation plane and-Trcontains the translation group of e, case (ii) fixes a point, O. The dual of the proof for case (i) proves this case. case (iiiNo point or line is left fixed by 17 Let P be the centre of a (P, t)-elation. If QEQ., is also the centre of an elation then, from lemma 16 , any homology centre incident with e. is the centre of an elation with axis E o Hence 11— contains an (R, Q )-elation for every REC. Since e., is not. fixed, -Tr cis o contains an (S ,m)-elation for all SE m, where m

Hence, as in case (iii) of Theorem 11 , 6) is desarguesian and t 60 contains the little projective group of 8) Assume that each elation centre (axis) has only one axis (centre). So every point. Dee, is the centre of a homology. If D has more than. one axis, then, by Results. H 32 is an elation

centre and-Wcontains a (D, O..-)-elation. Thus D can have only one

axis and this must pass through P. Two points CID Ee_cannot have the same axis since then, by Result I I , P would have two elation axes. Hence, for each point Dee , D#P, there exists exactly one line m through P, m such that Ti contains a (].,m)-homology. The point P cannot be a homology centre since any homology with

centre P would fix the axes through P but would mare their centres. Likewise, e, cannot be a homology axis. Thus every point of 9 must be the centre -of a perspectivity for exactly one axis, and every line of CS) must be an axis for exactly one centre. We define a mapping, 9 , of the points of G) onto the lines of (P and of the lines of Q onto the points of CP as follows:- For every point CE(?, C is the axis of C and for every line ee-e eg is the centre of e Since 9 is clearly an involutory mapping, in order to, show that 6%) is a correlation it is sufficient to show that if C e. , then ee E CO. If C is the centre of a (C, t)-elation, then C9 .e ,t9=C and e9S_Cg

If C is the centre of a (C,C9)-elation, A , CO •k then (ee )), is the centre of Hence ee is fixed by X But the only points which are fixed by A are the points of C9 . Thus &4€_CG. If C is the centre of a (C,C0)-homology, , then, as before,

61 egis fixed by /a . Since the only points fixed by,la are C

and the points of C 8 and since u*c, then 66.1)ECAD

Thus P is a polarity. The absolute pointa of Q- are the elation centres of . Since n#m2, we can now use Result 13. case (a) n even The elation centres must all lie on some fixed line m. which

contradicts our ar'sumption that no point or line of (9 is left fixed b5i-Tr-

cas2_1121 n odd

The elation centres must be such that no three are collinear. But if P and Q are any two centres and if A is an elation with centre P, then P, Q, QA are collinear.

Thus our assumption that there was only one elation centre (axis)

on each elation axis (centre) must be false and the theorem is proved.

Theorem 13 Let be a finite projective plane of order n and

, let--Ti- be a collineation group of (3) such that every point PE& 521 7 is the centre of a perspectivity Then either

(i) fixes a line of . Then& is a translation

plane and It contains the translation group of (P

(ii) 71 fixes a line of Then the conclusions are the dual of the conclusions for case (i) (iii) iT fixes no point or line of Then, if n4 m2, is desarguesian and contains the little projective

group of (P

Proof (i) 1 fixes a line, Qco Every affine point is the centre of a homology with axis clop

62 Hence, by corollary 2 of Result 11, e is a translation plane and 11 contains the translation group of . case (ii) --T1fixes a point, 0. We first note that the axis of every point, except 0, must pass through 0. If a line through, 0 has three non-collinear centres then, as in case (c) of Theorem 11, Jr contains a (P, e..)-elation for every Pr e. Since _ is not fixed, 7rcontains a (Q,m)-elation for every (lErn, m *e. Also, since these elationa are connected

;14d = Qi h 1 for all e., (4.m, P4:0, Q#0. Thus, from lemma 13, e is (0, )-transitive and (0,m)-transitive i.e. is the dual of a transitive plane and-7 contains the dual of the translation group of ?

If a line a. has more than one homology centre, but has no three non-collinear centres then, as in case (b) of Theorem 11, is (0,m)-transitive where m is the line of centres of Thus, since m is not fixed, the, theorem is proved.

If a line has more than one elation centre then, by lemma 16, TT- contains a (P (L ) -el ati on for all P Once again, since is not fixed, the theorem is proved. Since there are, hi-n centres other than 0 and only n+ 1 lines

through 0, one of the three cases considered above must occur and the theorem is proved. case n4 M2' and no point or line fixed byll— Remark 1 If an elation axis has more than one centre, the theorem is proved. From lemma 16, if Q. is an elation axis with centres A and B, thenlr contains a (C,e-)-elation for all C se.. Since e_ is 63 not fixed-Tralso contains a (Q,m)-elation for all Qam. These

elation are connected. The axis Q- has more than one centre,

the centre e n m has more than one axial andffi-does not fix a point or line. Hence by Theorems 4 and 5, a connected set must be a subplane of Thus every point of (3) is the centre of an elation in ?and the theorem is proved. Remark 2 If a homology axis has, three non-collinear centres, the theorem is proved.

From Result 11, any homology axis with three non-collinear centres is en elation axis with more than one centre.

Remark 3 If an elation centre is incident with a non-axis, the theorem is proved.

Let P be the centre of a (P l e„)-elation. If e„ has another elation centre then Remark 1 proves, the theorem. If we assume that P is the only centre of e_ then every point e.. C #P must be the centre of a homology with axis passing through P. If one of the lines through P is a non-axis then some axis, m, through P must have more than one homology centre

and, by Result 11,1i-contains a. (P,m)-elation. If any centre on. in has an axis other than e. , then m has three non-collinear centres and Remark 2 proves the theorem. Thus, H contains a (R, )-homology for all R E m, R.0 and by corollary 1 of Result 11, (53 is (P, e_.)-transitive.

Since P is not fixed, IS) contains another elation centre P / and, hence, every point of the line=-e PP is an elation centre. Since C., is not fixed, there exists an elation centre, // 0, such that o g The axis of 0 must meet e in an elation 64 centre and Remark 1 now proves the theorem.

Now let P be. the centre of a. (P,)-elation and. assume that for

any point S E e there exists a unique line m through P such -that m is the axis of S. Let Q E be incident with a line which is

not a perspectivity axis. (If such a line does not exist, then the conditions of the previous theorem are satisfied). Let the axis of q be m . Since the axis of every point of m must

pass through Q, there exists a line through Q, which is E1.11 / axis for two points of m / . Hence, by Result 11, e is an elation axis.

case (1)

Let R be any point of e , Rhp, R*Q. Then the homology with centre R fixes but moves mi. Thus e., has three non- collinear centres and Remark 2 proves the theorem.

case (ii)

If a point A E el has an axis e* # mI , then e., has three non-collinear centres and. Remark 2 proves, the theorem. Thus every

point of C. has m/ as an axis and, by Result 11, IT contains a

( gA m, m )-elation. Hence, by Result 2, Tr contains a (Ps mi )- elation, and Remark 1 proves the theorem.

Thus, if every point of (i) is the centre of a perspectivity, then.

every line is the axis of a perspectivity. If n* M2 then 6:3

is desarguesian, and iTcontains the little projective group of 6") . 65 6 ELATIONS OF FINITE PROJECTIVE SPACES OF ODD ORDER.

We now extend some of the results of § 4 to finite projective

spaces of dimension greater than two. Most of our results or e

restricted to spaces of odd order.

Throughout this section we shall use the word centre (axis) for

the centre (axis) of an elation. in IT.

Lemma 17, Let Ilk be a finite projective space and iet-n- be a collineation group of Hk. contains, a. (P,Hk_i )-elation t)(

and a (Q,Kk_i)-elation 3 such that PE ICk_a but Q then it contains a (P,Kk_1)-elation.

Proof The required elation is p

Lemma 18 Let Hk be a finite projective space of order p and let TT be a colineation group containing some elations. If no subspace of Hk is left fixed by —T, thenlris transitive on the centres (axes) of Hk.

Proof Let P and Q be any two centres and let e= PQ. If and Q has an axis iCk_i such that H P has an axis Hk-1 k-1 then I to has an element of order p fixing P (or Q) and and no other point of P, . Thus, by Result 1, P and Q lie in the same orbit under -rre Hence, either-Wis transitive on. the or there exist centres P and Q such that every centres of Hk axis of P contains Q.

is not transitive on the centres of H and let Assume that k

H be the intersection. of all the axes. of P. Then mapsP onto s s such that Ark H any centre of Hk\ Hs. Let A be a centre in H s 66 for s ome otEir. Then there exists an element parsuch that

("(a = A0( and po( -1 maps P onto A. Since we are assuming that 11 is not transitive on the centres of Hk, there must exist a centre, R say, such that RITE% for all -r ZIT--

Let Ht be the complete subspace spanned by the image points of R under Then H is left fixed b-y 11-. t and, since Ht cHs, t k-1. This contradicts our assumption that no subspace of Hk is left fixed by iF. Hence: Tr is transitive on the centres of Hk. The dual argument proves that—Fr-is transitive on the axes. of Hk.

Definition If an axis contains more than one centre, we shall

call any line which joins two centres with a common axis an

axis line,

c Lemma 19 Let Hk be a finite projective space of order p , where p#2, and letiTbe a collineation group of Hk containing

some elations. If Hk -1 is the axis of an elation in-IT-and if e..

is an axis line then either (i) e Hk-1

or (ii) e A Hk-1 is a centre

Proof We shall assume that e, Lot P and Q be any two with common axis If P E H or QE H. centres on e k-1 k-1' then the lemma, is true for a . Suppose P I Hk-1 and Q01k_i. Suppose there exists a point TEHk.e Kk_i which is a centre with axis Hk-1. Then, by lemma 17, T is also a centre for ICk_i.

Thus every point of Hk_lA ICk...1 which is a centre for Hk-1 is also a centre for Kk_i. But, since Ti is transitive on the axes of Hk, 67 each axis must have the same number of contres. Hence, since

PalCk_i is a centre for which does not belong to Hk-1,

there exists a centre R of Hk-1 such that 14K.k-1° Let c< be a (P,Kk....1)-elation, let A be a (Q,Kk..1)-elation,..and

let e be a (R211k....1. )-elation. Let HE be the desarguesian plane

H and let ILE = RS. defined by PQR, let S = e A k-1 The collineations are all collinoations of H2.

Since 0( fixes pointwise and fixes P linewise, either oC, is

a (P, e_)-elation of H2 or 0 I on H2. But Rd. R. Thus c( is a (p, e )-elation of H2. Similarly, /8 is a (Q, ) -elation of

112 and ,?( is a (R,m)-elation of H2. .,(3.,0 fixes k_iA pointwise and. is a The group <0( k-2 H collineation group of H2 fixing S. We first consider <0(, e> as a collineation group of H2. By Result 17, Ka)?S>contains an involutory (5 ,PR)-homol ogy of H2 and <73,p contains an involutory (S,Q,R)-homology of 112. Hence, by Result 11. <-°(.43,1S.> contains an

(S ISR)-elation, & say, of 112.

We now consider the effect of E on Ilk. The subspace Hk-2 and the line m , miHk_2 mellk....1 are fixed pointwise by chi , S fixes H Hence, since the points of Hk-2 and m span Hk-1 k-1 pointwise and is a perspectivity of Hk with axis IT_ - Since the points of any line through S together with the points of Hk-2 span a hyperplane which contains S, at least p-+1 hyperplanes through S are fixed by 3 Thus cS is a (S,Hk_i)-elation of Hk.

Lemma 20 Let Hk be a finite projective space of odd order and 68 letTrbe a collineation group of lc containing some elations

such that no subspace of Hk is left fixed by-Tr. If ' an axis has more than one centre, then each centre has more than one axis.

Proof Let P and Q be any two centres with a common axis H. and let e = since is_ not fixed by-rf-- , there exists an axis, Ku_i say, r' eh that e Kk_1. Then, by lemma 19 contains a (T,Kk_a )-elation where T=12 n K. But since .e,/ either Pit K.k-1 or Q Kk-1. Hence by lemma 18,-ff-contains a (T1Hk_1)-elation. Thus T has more than one axis and so, since, - by lemma 18,71 is transitive on the centres of Ilk' the lemma is proved.

Lemma 21 Let Hk be a finite projective space of odd order and letlT be a collineation group of Hk containing some elations such that no subspace of Hk is left fixed by- Tr . Then either the line joining any two centres is an axis line or there is only one centre per axis.

Proof Let be an axis with centres P and Q. Let R Hk_1 be another centre and let Kk-1 be an axis of R. Let e PQ. If then, using lemma 17, the lemma is true for the Kk-1 2 lines PR and QR. So assume C.ITc-1°- We may also assume that both P4Kk.a and Q4 Kk_a., since if PEICk_i then, by lemma 17, contains a (P,Kk_i)-elation, /'\ say, and Q,QX are two centres of Hk_i such that neither belongs to Ick_i. = T and let m= TR. Then from lemma 19, T is a Let Q A Kk-l 2 be the desarguesian centre. Let Hk-2 = Kk-1 and let H 69 plane PQRT. If o< is a (P,Hk_1)-elation, /3 is a (Q'Bk-1)- elation and 2( is a (R,Hk_1)-elation, then, as in the proof of lemma 19, c<,g)2( are all elations of H2. If Mk_i is any axis of R such that TaMk.a, then, since-IT- contains a (R,Mk_1)-elation and a (T,Hk_1)-elation and since RitHk_it 7-contains a (T,Mk_1)-elation of Hk (see lemma 18). Thus any axis of A. which contains T is also any axis for• T. Hence, since T has, an axis,Hk_11 which does not contain R and since R and T have the same number of axes, R must have an axis, 1,.. say, such that be a ( RI'llk_1)-elation of 1 1 4T-c-1° Let A Hk and let m = Lic..3.A H2. Then A is an (RA-elation of H2 and 4m: Now consider Tr-H.z This collineation group fixes no point or line of H . R is the centre of an elation in Tr for more 2 H2 than one axis and is an axis for more than one centre. Hence, by lemma 1, T is transitive on the axes of H and, by Theorem 4, H2 2 the line joining any two centres of H2 is an axis of H2. Thus since PQ is an axis line, RP and RQ are axis lines and the lemma is proved. spy Lemma 22 Let Hk be a finite projective plsae of odd order and let-Tbe a collineation group of Hk containing some elations. If no subspace is left fixed by-rrand if P is a centre for more than one axis, then the only centre fixed by is P.

Proof We first prove the following remark. Remark If II is the subgroup of 1T which is generated by the elations in TT-, then no point of Hk is left fixed by 70 / ProofPro-; Suppose P is left fixed by H Then every elation

of have its axis passing through P. Since doesnot fix P, other points of Hk must also be fixed by Tr . In fact, .„/ I must fix every point of the orbit of P underTr

Let Hs be the complete subspace of Hk spanned by the images

of P. Since all of these points are fixed by every elation,

they must all be contained in a hyperplane and, thus, s k 1.

But Hs is invariant under , which contradicts our assumption that Tr does not fix a subspace of Hk. This proves the remark.

Proof of lemma 22. Let H be the intersection of all the axes of t P and suppose P and Q are two distinct centres in Ht. Let Q= PQ.

Since no subspace of Hk is left fixed:TT-contains an elation, pc say, such that Poc, 4. P. Let OC have centre R and axis. Hk-1,

and let T = Hk_i. Then, by lemma 19, T is a centre. But,

since Ht, TE Ht and any axis, Kic_i say, of P must contain. T.

Thus, since 1T-contains a. (P,If.k_i)-elation and a (T,Hk.a )-elation

such. that P4 T aKk_i, we have, by lemma 17, that any axis

of P is also an axis for T. But T has an axis, Hk-1, not passing through P. This is impossible since P and T must have the same number of axes. Hence P must be the only centre in Ht.

Corollary If H2 is any desarguesian plane in Hk which contains

P then P is the centre of a non-trivial elation of H2.

be a finite projective space of dimension k Theorem 14 Let Hk and order pc, where p 2. If -Tr is a collineation group of Hk containing some elations such that - 71 (i) 11 fixes no subspace of Hk (ii) an elation axis has more than one centre, then

TT- contains the little projective group of Hk .

Proof From lemmas 18 and 20, the line joining any two centres is an axis line.

Let A and B be any two centres, and let Q= AB. By lemma 22,1Tcontains an elation with centre A which moves B. Thus, since this elation fixes e, , the line joining any two centres

contains at least three centres.

Let A, B, C be, any three non-collinear centres and D and E

(D#E) be two centres such that A, D, B are collinear and A, E, C

are collinear. Then, from the proof of lemma 21, DE ABC is a

centre.

Thus the centres and axis-lines are the points and lines of a subspace of Hk. , This subspace must be left invariant by iln

and must, therefore, be Hk itself. Hence, by Result 19, 1T contains the little projective group of Ilk

Finally we note that, although we have assumed the order of 11,_ to be odd, lemmas 20, 21, 22 and Theorem 14 are all true

whenever lemma 19 holds.

72

REFERENCES

1 ANDRE, J.: Uber Perspektivitaten in endlichen projektiven Ebenen. Arch. Math. 6. 29-32 (1954). 2 ARTIN, E.: Geometric Algebra. New York 195?. 3 BAER, R.: Projectivities of finite projective planes. Amer. J. Math. 69, 653-684 (1947). Homogeneity of projective planes. Amer. J. Math. 64,. 137-152 (1942). 5 Polarities in finite projective planes. Bull. Amer. Math. Soc. 52, 77-93 (1946).

6 BARLOTTI, A.: Le possibili configurazioni del sistema delle coppie punto-retta (A,a) per cui un piano grafico risulta (Al a)-transitive. Boll. Un. Mat. Ital. 12, 212-226 (1957). 7 BRUCK, R.H. and RYSER, H.J.; The non-existence of certain finite projective planes. Canad. J. Math. 88-93 (1949). 8 DEMBOWSKI, H.P.: Verallgemeinerungen von Transitivitatsklassen endlicher projektiver Ebenen. Math. Z. 69, 59-89 (1958). 9 DICKSON, L.E.: Linear Groups. Dover. N.York (1958) 10 GLBASON, A. M.: Finite Fano planes. Amer.J.Math. 78, 797-807 (1956). 11 HARTLEY, R.W.: Determination of the ternary collineation groups whose coefficients lie in GF(211). Annals of Math. 27, 140-158 (1925-26). 12 HILBERT, D.: Grundlagen der Geometrie. 1st ed. Berlin 1899.

13 HUGHES, D.R.: A class of non-desarguesian projective planes. Canad. J. Math. 9, 378-388 (1957).

14 Collineations and generalised incidence matrices. Trans. Amer. Math. Soc. 86, 284-296 (1957).

15 LENZ, H.: Kleiner Desarguesscher Satz and Dualitat in projektiven Ebenen. Jber. dtsch. Math.-Ver. 57, 20-31 (1954). 73 16 LEVI, F.W.: Finite geometrical systems. Calcutta 1942. 17 MITCHELL, H.H.: Determination of the ordinary and modular ternary linear groups. Trans. Amer. Math. Soc. 12, 207-242 (1911). 18 NOUFANC, R.: Die Schnittpunktsatze des projektiven speziellen Funfecknetzes in ihrer Abhangigkeit voneinander. Math. Ann. 106, 755-795 (1932).

19 Alternativkorper and Satz vom vollstandigen Vierseit. Abh. math. Seminar Univ. Hamburg 9, 207-222 (1933).

20 OSTROM, T.G.: Double transitivity in finite projective planes. Canad. J. Math. 8, 563-567 (1956).

21 Transitivities in projective planes. Caned. J. Math. 9, 389-399 (1957).

22 A class of non-desarguesian affine planes. Trans. Amer. Math. Soc. 104, 483-487 (1962).

23 Semi-translation planes. Trans. Amer. Math. Soc. 111, 1-16 (1964).

24 PARKER, On collineations of symmetric designs. Proa. Amer. Math. Soc. 8, 350-351 (1957).

25 PIOKERT, G.: Projektive Ebenen. Heidelberg 1955. 26 PIPER, F. C.: Elations of finite projective planes. Math. Z. 82, 247-258 (1963). 27 SAN SOUCI, R.L.: Right alternative rings of characteristic two. Proc. Amer. Math. Soc. 6, 716-719 (1955). 28 SEGRE, B.: Ovals in a finite projective plane. Canad. J. Math. 7, 414-416 (1955).

29 u Sulle geometrie proiettive finite. Convegno internazionale Palermo 46-61 (1957).

30 VEBLEN, O. and BUSSEY, W. H.: Finite projective geometries. Trans. Amer. Math. Soc. 7, 241-259 (1906).

31 VEBLEN, 0. and YOUNG, J.W.: Projective geometry. 2nd ed. Boston 1916. 74

32 WAGNER, A. On perspectivities of finite projective planes. Math. Z. 71, 113-123 (1959). 33 IT On collineation groups of projective spaces 1. Math. Z. 76, 411-426 (1961). 75 NOTATION !ND DEFINITION OF CONNECTED SET

NOTATION

IfTr is a. collineation group of a projective planet? and if P is a point of C9 then: subgroup of 71 fixing P lorbit of P undorl = number of points onto which P is mapped by elements of

group of all (›K) e)-elations — "e_ DEFINITION

Two elation centres P and Q are said to be connected

if there is, a sequence of incident centres and axes joining

P to Q. "Connectedness" between two axes, or between a centre

and axis, is defined in the same way. The set of all centres and axes which are connected to a

given centre P is called a connected set. A connected set

which consists of a flag only is described as trivial. If a

connected set contains more than one centre and more than one axis, it is said to be a proper connected set. The characteristic of a connected set is equal to the order

of any elation whose centro lies in that set.

The degree of a connected set is equal to 12P,CI for any centre P of the set and any axis of the set with PEP. note, that the degree is always a power of the characteristic. PIPER, F. C. Math. Zeitschr. 82, 247-258 (1963)

Elations of finite projective planes By F. C. PIPER

§ 1. Introduction The purpose of this paper is to generalize the following recent result due to WAGNER [10]. Theorem A. Let 13 be a finite projective plane and let H be a collineation group of $ with the following properties: (i) For every point CE$ there exists an element AEH such that A is an elation with centre C. ' (ii) For every line 1E 43 there exists an element e gu such that e is an elation with axis 1. Then $ is desarguesian and H contains the little pro- jective group of $ as a subgroup. This theorem postulates that every point of j3 is the centre of an elation and every line of 13 is the axis of an elation. In § 3 we postulate only that every point is the centre of an elation. From this weaker assumption we show that either the conclusions of Theorem A follow or a point of 13 is left fixed by H. When a point is left fixed, $ is the dual of a translation plane and H contains the dual of a translation group as a subgroup. In § 4 we weaken our postulates further by assuming that every flag is left fixed by an elation. This new postulate allows either a point of 3 or a line of 3 to be left fixed by H. If a point is left fixed, the conclusion is the same as in § 3, while a line being fixed gives the dual result. When no point or line is fixed, $ must be desarguesian and, except for n=4, H contains the little projective group. When n=4, H either contains the little projective group or lI=AB or S,. That this situation actually occurs is shown in § 5.

§ 2. Definitions and previous results Projective plane and collineation are defined in the usual way, see for example PICKERT [7]. It is well known that the collineations of a plane form a group which we call the collineation group of 3. Since collineation groups may be regarded as permutation groups on the points (or lines) of a plane, we shall adopt the following conventions. If H is a permutation group on a set S and if P E S , then we denote the subgroup of H fixing P by H. The set of points Per, for all nE17, is called the orbit (or transitive class) of P. We note that !HI = I1I •I orbit of PI.

248 F. C. PIPER:

A collineation of order 2 is an involution. A (P, l)-perspectivity is a collineation fixing all the points of a line, 1, and all the lines through a point, P. P is called the centre of the perspec- tivity and 1 its axis. If P El the perspectivity is an elation. The term elation will normally be understood to mean an elation +1. The group generated by all the elations of P is called the little projective group. An affine plane is a projective plane with a distinguished line lco . An elation of the affine plane $* with axis /00 is a translation. The group generated by all the translations of a plane is the translation group. If the translation group of J3* is transitive on the affine points of r, then 3* will be called a translation plane. A projective plane will be said to be (C,1)-transitive if for the fixed point C, the fixed line 1 and any pair of points Q and Q' with Q, Q'El; Q, Q' +C; C Q = C Q', there exists a (C, l)-perspectivity, a, with Q o: = Q'. For a definition of a desarguesian plane see, for example, PICKERT [7]. We note here, however, that a finite plane is desarguesian if, and only .if, it may be coordinatized by elements of a field. If $ is a finite projective plane of even order n, then a set of n +2 point such that no three are collinear will be called an oval. If $ is a desarguesian plane of even order n and V is a conic in $, then the n 1 tangents to' are concurrent at a point called the nucleus of W. (For a definition of conic and a proof of the above property see SEGRE [9].) We now list some well known results which we shall repeatedly use: Result 1. (See GLEASON [3].) 43 is a finite projective plane and H a collineation group of 43. If for two distinct points X, Y, (X El, Y El) H contains non-trivial (X, 1) and (Y, 1)- elations, then all elations in H with axis 1 have the same prime order. Result 2. (This is a simple extension to Lemma 1.7, GLEASON [3]. Gleason's proof essentially still applies.) G is a permutation group on a set S. T is a subset of S such that, given any A E T, there exists an element gEG of given order p such that g fixes A and no other element of S. Then T is contained in an orbit of G. Result 3. (See DEMBOWSKI [2], HUGHES [5], PARKER [6].) Let l be a finite projective plane and let 11 be any collineation group of 3. Then the number of transitive classes into which 11 splits the points of $ is equal to the number of transitive classes of lines. Result 4. (See GLEASON [3].) Notation. Gx,1 = group of all (X, l)-elations; G1 = U Gx,1. XE1 is a finite projective plane of order n. Suppose for some line, 1, all the groups Gx,1 have the same order h> 1, then G1 is transitive on the points of F13\l. Elations of finite projective planes 249

Corollary. If H is transitive on the points of l and if I Gx,11 >1 for some point XE/, then GI is transitive on the points of q3v and q3 is a translation plane. Result 5. (Unpublished; due to HUGHES.) 13* is a finite affine plane and 0 is a point of la,. If for every point P P +0, there is a non-trivial (P, 7J-translation in H, then H contains a non- trivial (0, /c.)-translation. As this result is unpublished we sketch a proof. Let pi (1=1, 2, ..., n) be the centres of non-trivial translations oci, 1, 2, ..., it). Let m be any line through m+/„.. There are n —1 possible positions for m oci and n different oci. Hence there exist cci, ak, j+k, such that m ai=m ak. Thus cci ock-i, +1, leaves lc° fixed pointwise and fixes m, i.e. oci-1, is a (Pn+i, 4J-translation. Result 6. (See BAER [1].) Let $ be a projective plane of order n, and let oc be an involutory col- lineation of 3. Then either cc is a perspectivity or n is a square and oc fixes elementwise a subplane of $ of order rn. Further if n is even and cc is a perspectivity, then cc is an elation. Result 7. (This is more precise statement of Theorem 1, WAGNER [10].) is a finite projective plane and H a collineation group of $. If /E$ is the axis of an elation in H and if ClE is the centre of an elation in H, then H contains a (C, /)-elation.

§ 3. Every point is the centre of an elation Theorem 1. 13 is a finite projective plane and H a collineation grout of $. For every point of j3 there exists an elation in H with that point as centre. Then either (i) H fixes a point of $. Then $ is the dual of a translation plane and 11 contains the dual of the translation group as a subgroup. Also, H is transitive on flags not containing the fixed point. (ii) /I does not fix a point of $. Then $ is desarguesian and H contains the little projective group as a subgroup. Proof. We first prove the following lemma: Lemma 1. If every point of $ is the centre of an elation in /I, then all the elations in I7 have the same prime order. Proof. Let C be any point of l3. Let C be the centre of an elation with axis l and p* ne order p. Consider mother point D (DE1) with axis m. Case 1. I, m=C From Result 7, there exists a (C, m)-elation. From the dual Result 1, this (C, m)-elation has order p and so, from Result 1, the (D, m)-elation has order p.

250 F. C. PIPER:

Case 2. Ir--,nt=:C', C'* C. By hypothesis C' is a centre so, from Result 7, there exists a (C', m)-elation and a (C', /)-elation. Repeated use of Result 1 and its dual give: the order of the (D, m)-elation = the order of the (C', m)-elation = the order of the (C', /)-elation = the order of the (C, /)-elation = p. Thus for any point DE '43v the order of elations centre D is p. But any point PEI is the centre of an elation, axis 1, which must have order p. If PE1 is the centre of an elation with axis in, (m+1), then the order of (P, in)- elation = the order of (P, /)-elation = p. Thus all elations of $ have the same prime order p. Theorem 1 is now proved by the following sequence of remarks. Remark 1.1. Let l be any line of $. If for every point PEI there exists a line in through P (m+1) such that m is the axis of am elation, then H is transitive on the points of 1. In particular, H is transitive on the points of non-axes. Proof. From Result 7 every point PE/ is the centre of an elation whose axis is not /. Consequently, for every point PE/ there exists an elation of order p in Hi which fixes P and no other point of 1. But Hi is a permutation group on the points of 1. Hence, from Result 2, 111 is transitive on the points of I. Remark 1.2. Either H is transitive on the points of $ or H fixes a point of $. Proof. From Remark 1.1, 17 is transitive on the points of any line / unless there exists a point PEI such that / is the only axis through P. Hence either H is transitive on the points of l or there exists a point P as above. Assume H is not transitive on the points of 3, i.e. that there exists a point P with only one axis, 1, through it. Since H is transitive on the points of non-axes and since every line through P, except of course 1, is a non- axis, H maps P onto any point of q3\l. Let Q be any point of 1, Q + P. If Q tr=-- R, Rffl, then there exists a collineation, cc, of H such that Pcc=R, and or. gri- maps P onto Q. Hence, since H is not transitive on the points of P, there exists a point OE/ such that 0 vEl for all nEfl. Let D be any point of $\1. D is the centre of a (D, m)-elation, cc. Since 0 ccEDO and since 0 ocEl, 0 must be left fixed by cc and m must pass through 0. Thus every point DE $\/ is the centre of a (D, OD)-elation and can have no other axis. Hence D has the same property as P and, repeating the above argument with D instead of P, it follows that H is transitive on the points of q3\0. Hence 0 is left fixed by 17 and, if X is any point of q3\o, all elations with centre X have axis X0. We are now in a position to complete the proof of Theorem 1. Case (i). 17 fixes a point, 0. Elations of finite projective planes 251

From the proof of Remark 1.2, it follows that H is transitive on the points of $\0. Hence the number of transitive classes of points of $ is two and it follows, from Result 3, that the number of transitive classes of lines is also two. H cannot map a line through 0 onto a line not through 0, so H must be transitive on the lines through 0 and also on the lines not through 0. Hence, from the dual of the corollary to Result 4, $ is the dual of a translation plane. The number of lines through 0 is n + 1. The number of lines not through 0 is 0. But (0, n +1) =1. So, for any two lines 1, m, through 0 and any two lines 11 , m,, not through 0, there exists a collineation nEll such that la=m and 1, n=mi , i.e. H is transitive on the flags of $ not through 0. Case (ii). H does not fix a point of $. From Remark 1.2 we know that H is transitive on the points of $. But it has been shown by WAGNER [10] that if H is transitive on the points of 3 and if H contains a perspectivity, then $ is desarguesian and H contains the little projective group.

§ 4. Every flag is fixed by an elation Theorem 2. 43 is a finite projective plane of order n and H is a collineation group of q3. For every flag of $ there exists an elation in H fixing that flag. Then either H fixes a point of $. Then ',)3 is the dual of a translation plane and H contains the dual of a translation group. H fixes a line of $. Then $ is a translation plane and H contains the translation group of $. (iii) H does not fix a point or a line of $. Then $ is desarguesian. If n + 4, then H contains the little projective group, while if n= 4 H either contains the little projective group or is isomorphic to A6 or S6. Proof. We note that if the flag (P, 1) is fixed by an elation, then either P is a centre or 1 is an axis. Hence every point on a non-axis is a centre and every line through a non-centre is an axis. Lemma 2. If every flag of $ is left fixed by an elation in H, then all elations in H have the same prime order. Proof. Suppose H contains elations of order pi (i = 1, 2, ..., k) pi+pi if i +i. From Results 1 and 7 we assert that no axis can contain centres of different orders. Let I be a non-axis and let ni be the number of pi centreson 1(E ni = n 1) . i=1 Without loss of generality we may assume (j= 2, 3, ..., k). Let in be an axis through a prcentre on 1. Clearly in contains only /51- centres and non-centres. Let ki be the number of pr centres on in and let k2 be the number of non-centres on in (k1+1e2=n+ Let frInt=1-1, and let Q be a non-centre on m. Each line joining Q to a pi-centre on 1 is a praxis. Hence there are ni praxes through Q. Each

252 F. C. PIPER: p1 elation with axis m is a permutation on the n, —1 praxes through Q, other than In. Let cci be a (Pi , m)-elation (i =1, 2, ..., k1) and let l' be a praxis through Q, *m. If l' eci=l'ai for i=j, then cei tx71 is a (Q, m)-elation. But Q is a non- centre. This contradiction shows Vai +ra j for i+j. Hence:

(i) ki < ni —1. Let R be a pi-centre on 1, j *1. The lines joining R to the k2 non-centres on in are praxes. Hence there are k2 praxes through R. Let mi (1 =1, 2, , k2 ) be these axes and let Ai be a (R, mi)-elation. Each Ai is a permutation on the ni p1-centres on 1. If P Ai= P Ai for i +j, then Ai AT' is a (R,1)-elation. But 1 is a non-axis. This contradiction shows (ii) k,< n1. k From (i) and (ii), n +1 =k,+ k2< 2n, —1. But 2 n1;--- ni= n 1(2n,= i=1 n if k=2 and n1=-- n2 ). Thus n + 1 = k, k,< — 1 n, NSthich is im- possible. Hence all elations in H must have the same prime order. We now prove Theorem 2: — Case (i) H fixes a point of $. Let the point be 0. Clearly every axis passes through 0. Take any point P +0 and any line 1 such that PE/, 0E1. P is the centre of a (P, 0 P)-elation. Any line 1 through 0 is the axis for n different centres. So, from Result 5, 1 is an axis for n centres i.e. there exists an (0,1)-elation. Hence every point of 4; is the centre of an elation and case (i), Theorem 1 applies. Case (ii). The dual argument establishes this case. Case (iii). H fixes no point and no line. We assume that there is a point of 13 which is not the centre of an elation in H, and a line which is not an axis. If this is not so, then the conditions of Theorem 1, or its dual, are satisfied. Theorem 2 is proved by the following sequence of remarks. Remarks 2.1. 11 is transitive on the points of non-axes and the lines of non- centres. Proof. Consider a non-axis 1. For every point Pa, HI contains an elation of prime order p which fixes P and no other point of 1. Hence, from Result 2, Hi is transitive on the points of 1. The dual of the above argument completes the proof of Remark 2.1. Remark 2.2. Every point of 43 has at least two axes through it. Proof. Remark 2.2. is certainly true for non-centres. Suppose there exists a point PE13 with n non-axes through it. Let 1 be the axis of P. In view of Remark 2.1, we can map P onto any point of 43v. Hence, as in the proof of Theorem 1, either H is transitive on the points of 43 or there is a point Elations of finite projective planes 253

QE/ such that Q nEl for all nEll. The first alternative contradicts our as- sumption that there exists a non-centre, so the second alternative must hold. Consider any point R E \ /. The flag (R, RP) is fixed by an elation, RP is not an axis, so R must be the centre of an elation a. QaERQ and QaEl. Hence Q a = Q i.e. Every point of q3 \ / is the centre of an elation with axis passing through Q, and can have no other axis. But every line not through Q will contain some point ofc3\ 1, so, from Result 7, there can be no axis not passing through Q. This means that Q is a fixed point and contradicts our assumption. Thus every point has at least two axes through it. Remark 2.3. H is transitive on centres and on axes. Hence every centre contains the same number of non-axes and every axis the same number of non-centres. Proof. II is transitive on the centres of non-axes. Consider an axis 1. From Remark 2.2 and Result 7, it follows that every centre PE/ is the centre of an elation whose axis is not 1. This elation fixes P and no other point of 1. So, from Result 2, the centres on 1 lie in an orbit of HI . They must, in fact, be an orbit since it is impossible to map a centre onto a non centre. Hence II is transitive on the centres of q3 and, by a dual argument, is also transitive on the axes of 3. Remark 2.4. II is transitive on non-centres and H is transitive on non-axes. Proof. Consider an axis 1 and a point PE/. An elation oc centre P axis m (m+1) permutes the points of 1. Since the order of the elation is 1), the points of q3\ 1 lie in orbits under of length divisible by p. Thus the number of non- centres on an axis must be divisible by p. Let the number of non-centres on each axis be kp. Any non-centre has n + 1 axes through it. Each axis has k p non-centres, so the total number of non-centres is (n +1) kp -n. Assume there exist at least 2 orbits of non-centres. Let P and Q lie in different orbits and let SQ be the number of points in the orbit containing Q which lie on the line P Q. Every line through P is an axis and is in the orbit of the line P Q. So every line through P must contain SQ points of the orbit of Q. Hence I orbit of Q I = (n + 1) S. Repeating for P gives I orbit of P1 = + I) S. The total number of non centres is equal to the sum of the orbit lengths i.e. (n +1) kp—n=E (n. +1) S. This is impossible since n +1 1r(n+1) kp —n. Hence, all non centres lie in one orbit. The dual argument completes the proof of Remark 2.4. Remark 2.5. If there are kp non-centres on an axis, then the number of non-axes through a centre is — k p ' Proof. Let C be a centre and let t be the number of axes through it. Each axis contains kp non-centres and each non-centre is on one of these axes. So

254 F. C. PIPER: tkp = number of non-centres in 13= (n +1) k p -n, i.e. t z=n +1- Hence k p the number of non-axes through a centre is n +1 -t=--L-1 kp ' Remark 2.6. Every point of 13 must be the centre of an elation except, possibly, when n=4. Proof. Either kp1 for all centres PEl. But every element of GPI has order p, hence h=p. Gi= U Gp,/ so I GII =(n+1-kp)(p"-1)+1. But every element of G1 has PEI order p, so 1 G11 = ps where 8> a. Hence (i) (n + - k P) (25" — 1 ) ± =p°. Let m be any line through a centre, 0, on 1 and let cr be the subgroup of GI fixing m. Then I Gil G1 . orbit of m under Gi l . :But Gr hence orbit of m under G,I=po-.. Thus the n lines like vt, through 0, distinct from 1 split into orbit of length p13-" i.e. pp--.1 n. Let apfl= np" (a 1), then, from (i) (ii) - n + kp.+1+ a p = pP Case (i). a=1: Substituting np-= pfs in (ii) gives n = - kp.+1 kp = + k (1 — But n> kp, hence p- k p (1 -p.)> hp i.e. pm - O. This is im- possible since p> 1 and k> 0. Case (ii). a> 1: Substituting n=ag-' in (ii) gives a9 - a pfl -.+ - kp.+1+ kp =po, and hence po (a —I) + apo--. - k pm- fi - hp. ft—cc But kp < Vn, hence hp- ilapo-- = p 2 . ft—a Thus p° (a - 1) - F apo-. ya p 2 (p -1), i.e. P—a aP13 — - P-5-ag — + P 2 (PM 1) /3—CE a (P# P'3-a) — 1/a P 2 (Pct 1) —I— PS—cc (Pa 1)5°, Elations of finite projective planes 255

i.e.

ae-m (Pa 4) Pfl cc (Pm + P fi-a (Pa i) 0 • But pa — I> 0, so that P-cc (i) apfl—. p 2 -43-"0. Put X2= apP--. in (i) where X> 0; then X2 — X - - X 2 0 , a whence --1a—.7( S0, i.e. x — 5 . Hence (ii) X a — a- 1 Putting a =2 in (ii) gives 2, X2 4 and hence pft—as 2. But /3> a, and so pfl-->1. Thus pf3"=2 and n -= aP13-a-= 4. If, however, a z 3 then it follows from (i) that X z 3, and from (ii) that X_<-I which is impossible. Hence, unless n=4, all points are centres. The proof of Theorem 2 is now complete except for the case rt•=4. There is only one plane of this order, and, as this is known to be desarguesian, our last problem is to discuss its collineation groups. Remark 2.7. If a point of is not the centre of an elation'), then n=4, the points which are not centres form an oval, the lines which are not axes form the dual of an oval and 11..-" A, or S6 . Proof. As before, kp is the number of non-centres on an axis, kp =2 or 4. Suppose kp =, 4. Then the number of non-axes through a centre is-iv, =1. But, by the dual of Result 5, this is impossible. Hence we will assume kp =2. The number of centres is 15. Hence Ill" =45. I Hc where Ho is the sub- group of H fixing a centre, C. H is transitive on the axes through C and on the non axes through C. Hence If/1=15.3.2. IIIc,t,rI where .17c ,,,T is the subgroup fixing a centre, C, an axis, 1, through it and a non-axis, 1, through it. Hc,/,-E contains three elations of order 2 with centre C and different axes. These, together with the indentity, are a four group. Hence Illc,0-[= 4h and I HI =-- 360 h (where h is some constant). There are six non-centres and no three are collinear i.e. the non-centres form an oval. Since H is transitive on the non-centres, H maps the oval onto itself. Suppose there exists a collineation nE.//such that a fixes the oval pointwise. Four points of the oval form a non degenerate quadrangle, so a fixes a 7 point

1) The proof that this situation actually occurs is found in § 5.

256 F. C. PIPER: subplane of 13. This subplane cannot contain any other point of the oval because a 7 point plane does not contain five points such that no three are collinear. Hence, since this subplane is maximal in the plane of order four, H must fix the whole plane and must be faithful on the oval. But the only groups of order 360 h faithful on six letters are A,, when the order is 360, and S6, when the order is 720. This proves Remark 2.7. and completes the proof of Theorem 2.

§ 5. Collineation groups of 14 In this section we prove four remarks which completely answer two ques- tions posed by Theorem 2. They are; (i) Does 434 have a collineation group isomorphic to A, or S,?: and (ii) If 17.' A, or S, is a collineation group of 134, is every flag of 434 left fixed by an elation in H? The fact that A6 can be represented on 134 was shown by HART- LEY [4]. Remark 3.1. Given any oval of 43, there exists a collineation group Hz S, mapping the oval onto itself. Proof. Let H be the collineation group fixing an oval (such a group cer- tainly exists, though it may consist of the identity alone). Let A, B, C, D, E, F be the points of the oval. 43 is desarguesian and can be coordinatized by elements of GF(4). Choose A = (1, 0, 0), B= (0, 1, 0), C=(0, 0, 1), D=(1, 1, 1). The two remaining points of the oval must be (w, cot, 1) and (0)2, w, 1). BCDEF is therefore the conic yz= x2 and A is its nucleus 2). There is only one non-trivial automorphism, a, of G.F(4). a maps 0—>0, 1—>1, cue—w, i.e. a fixes A, B, C, D and interchanges E and F. We can choose any four points of the oval as our triangle of reference and unit point. So, for any four points of the oval, there exists an element of H fixing those four points and interchanging the other other two Hence II= S,. Corollary. There exists a collineation group ^A6 which maps an oval onto itself. Remark 3.2. Every collineation group 17-__•-• A, fixes an oval and contains only linear transformations. Proof. The derived subgroup of 17 consists only of linear transformations (since the automorphism group of a finite field is cyclic:). H is simple and non-abelian. Hence H coincides with its derived group and H contains only linear transformations. A similar argument shows that H must contain only transformations whose determinant is one i.e. H is a subgroup of the little

2) The relationship between ovals and conics in desarguesian planes is discussed by SEGRE [81 Elations of finite projective planes 257 projective group, L, of $.

IHI =360 IL I =- 21.20.16.3 IHI =360 I Lpl=20.16.3. But 3601-20.16.3. so H cannot fix a point of $. The smallest orbit of H must have length at least six because A, cannot be faithfully represented on less than six letters. Hence I smallest orbit of HI = 6, 7, 8, 9, 10 or 21. But 71'360, hence I smallest orbit of HI + 7, +21. If I smallest orbit of HI =8 or 10, the remaining points must form one orbit of length 13 or 11 respectively. But neither of these numbers divides 360, so now we have only to eliminate 9. Let P be a point of an orbit of length 9. 1141=40, I 41=20.16.3. Hp is a permutation of the eight remaining points of this orbit. Hence I smallest orbit of Hp! =2, 3, 4, 8. Now 34-40, so !smallest orbit of 17- pi = 2, 4, 8. Let Q be a point of the smallest orbit of Hp . If I smallest orbit of Hpl = 8, IHP42I = 5 and 14Q = 16. 3 . But 51'48 so I smallest orbit of Hp' =8. Similarly it is impossible for I the smallest orbit of //pi to be 2 or 4 and there can be no orbit of length 9. Hence I smallest orbit of HI = 6. Suppose P, Q, R are 3 collinear points of this orbit. H 0 is certainly transitive on the remaining points, so all six points must be collinear. This is impossible. Hence no 3 points of the orbit are collinear i.e. the orbit is an oval. Remarks 3.3. If 11-' A, then every flag of Fp is left fixed by an elation. Proof. Let 93' be the set of points of the plane not on the oval fixed by H. Since we have shown that H cannot have an orbit of length 7 or 9, H must split the points of J3 into 2 transitive classes. I HI =360 so H contains an element of order 2. This must either be an elation or fix a 7 point plane (see Result 6). But H contains only linear transformations and cannot have an element fixing a subplane. Hence H contains an elation. Suppose a point A of the oval is the centre of an elation with axis 1. Let B be another point of the oval B (El. Then Bcc+ B but BG(EAB i.e. Bac is not in the oval. This is impossible, so there exists a point, PE 43', which is the centre of an elation. H is transitive on the points of $', so every point of $' is the centre of an elation. Every flag of $ with a point of j3' is fixed by the elation with that point as centre. Let A be a point of the oval, let 1 be any line through A and 1 cut the oval again at B i IIA,Bi =12. Hence 114,B contains an elation fixing the flag (A,1). This proves Remark 3.3. Remark 3.4. If Ths-_•--S, then H fixes an oval and every flag of $ is left fixed by an elation. Proof. Let //' be the subgroup of H isomorphic to A6 . H' is normal in H and fixes a unique oval. Hence H fixes the oval fixed by H'. /7'< H, so every flag of $ is left fixed by an elation in H. Mathematische Zeitschrif t. Bd. 82 18 258 F. C. PIPER: Elations of finite projective planes

Finally, we observe that if every flag is left fixed by an elation and if H is generated by these elations, then //f-_—_-A6 . I wish to express my gratitude to Dr. A. WAGNER for suggesting the problem and for his valuable assistance with this paper.

References [1] BAER, R.: Projectivities with fixed points on everyline of the plane. Bull. Amer. Math. Soc. 52, 273-286 (1946). [2] DEMBOWSKI, H. P.: Verallgemeinerungen von Transitivitatsklassen endlicher pro- jektiver Ebenen. Math. Z. 69, 59-89 (1958). [3] GLEASON, A. M.: Finite Fano planes. Amer. J. Math. 78, 797-807 (1956). [4] HARTLEY, R. W.: Determination of the ternary collineation groups whose coefficients lie in GF (2"). Annals of Math. 27, 140-158 (1925/26). [5] HUGHES, D. R.: Collineations and generalized incidence matrices. Trans. Amer. Math. Soc. 86, 284-296 (1957). [6] PARKER, E. T.: On collineations of symmetric designs. Proc. Amer. Math. Soc. 8, 350-351 (1957). [7] PICKERT, G.: Projektive Ebenen. Heidelberg 1955. [8] SEGRE, B.: Ovals in a finite projective plane. Canad. J. Math. 7, 414-416 (1955). [9] — Sulle geometrie proiettive finite. Convegno internazionale Palermo 46-61 (1957). [10] WAGNER, A.: On perspectivities of finite projective planes. Math. Z. 71, 113-123 (1959)• Mathematics Department, Imperial College, London (Received March 9, 1963)