on flag collineations of finite projective planes

Journal of Geometry Voi.28 (1987)

0047-2468/87/020117-1151.50+0.20/0 �9 Birkhguser Verlag, Basel

ON FLAG COLLINEATIONS OF FINITE PROJECTIVE PLANES

Adilson Goncalves and Chat Yin Ho*

This paper studies collineation groups of a finite projective plane containing flag collineations. Among other results, a characterization of a finite Desarguesian projective plane is given.

I. INTRODUCTION

A flag collineation of a projective plane is a collineation, whose fixed

points and lines form an incident point - line pair. In a Desarguesian

projective plane, there are two kinds of collineations induced from the

unipotent linear transformations of the underlying vector space. An elation

(resp. flag collineation) corresponds to a transformation whose minimal

polynomial is (x - 1) 2 (resp. (x - i)3). Questions involving elations have

been widely studied. On the other hand~ little is known in the case of flag

eollineations for projective planes in general.

The main purpose of this paper is to prove the following theorem using the

recently-obtained classification of finite simple groups.

THEOREM. Let ~ be a finite projective plane of odd order n, G be a

co!lineation group of ~ containing non-trivial perspectivity. Suppose G has a

subgroup of order n such that non-trivial elements of this subgroup are flag

collineations with a common fixed point and a common fixed line. Then one of

the following holds.

(i) G leaves invariant a point and a line.

(ii) G leaves invariant a line but no points and ~ is a translation plane.

(*) Partially supported by grants from CNPq do Brasil and NSERC of Canada~

118 Goncaives and Ho

(iii) G leaves invariant a point but no lines and ~ is a dual translation

plane.

(iv) G doesn't leave invariant any point or line and ~ is Desarguesian.

Other results on flag collineations are present in section 3. Results in 3.4~

3.5 concerning the fixed point line substructure of collineation group

containing flag collineations might be of independent interest.

2. DEFINITIONS AND SOME KNOWN RESULTS

In this paper ~ = (P,L) will be a finite projective plane of order n and G

will be a collineation group of ~~

For H < G, set P(H) = {P E PIP h = P for all h e H}, L(H) = {% C Lji h = ~ for

all h c H}, and Fix(H) = (P(H),L(G))~ For x c P and x 6 L let

[X] = {% 6 LIX s %} and (x) = {P c PIP c x}o

We call a collineation g of ~ a flag if Fix(g) is an incident point-line pairs

planar if Fix(g) is a subplane; a generalized perspectivit~ if

F(G) < {A} ~ (a) and L(G) < [A] ~ {a} for some A s P and a s L. A generalized

perspectivity g is a generalized homology (resp. elation) if A ~ (a) (resp.

A 6 (a)). The unique fixed point (resp. line) of a flag collineation g is

called the center (resp. axis) of g. A generalized homology is triangular if

there are exactly 2 fixed points in its axis. We also apply these terms to

groups of eollineations of E. Thus, for example, a flag group with center A

and axis a is a group of collineations whose non-trivial elements are flag

collineations with common center A and common axis a. The notation F~(A~a)

means a flag group with center A and axis a.

We say that G acts irredueibl~ on E if G does not leave invariant any polnt~

line, triangle. An irreducible collineation group is strongly irreducible if

it does not leave invariant any proper suhplane.

Other definitions in group theory and geometry are standard and can be found

in [I], [2], and [6]. All objects considered in this paper are finite.

For the convenience of the reader, some of the known results are co!leeted in

the following.

THEOREM. Let E be a finite projective plane and G he a collineation group of

containing a non-trivial perspeetivity. Suppose G acts strongly irreducibly

on ~. Then there is a unique minimal normal subgroup M of G such that

Goncalves and Ho 119

Fix(M) = (~,~), and M is either an elementary abelian subgroup of order 9

satisfying CG(M ) = M or a non-abelian simple group with CG(M) = i ([3]).

Furthermore, the following holds.

(i) If M ~ PSL(3,q), then ~ ~ PG(2,q) except possibly when

M ~ PSL(3,2) ~ PSL(2,7) ([4]).

(ii) If G contains a non-trivial elation and M is isomorphic to a simple

Chevalley group of odd characteristic of rank I, then ~ is isomorphic

to PG(2,q2); PG(2,4); PG(2,2), respectively, when M is isomorphic to

esu(3,q); PSL(2,9); PSL(2,7) ([5]).

(iii) If M is isomorphic to a sporadic simple group, then M ~ J2' the Janko

group of order 27.e3.52.7 ([II]).

(iv) If M is isomorphic to a simple Chevalley group, normal or twisted of

rank m, then m < 2 ([12]).

(v) If M ~ PSU(3,q) and the involutions in M are homologies with distinct

centers and distinct axes, and (q,n) ~ i then ~ ~ PG(2,q 2) ([8]).

(vi) If M is isomorphic to an alternating group on r letters, then r < 7

(Hering, unpublished).

(vii) If M is a simple Chevalley group, normal or twisted of rank 2, then

M ] PSL(3,q) and ~ % PG(2,q) (Walker, unpublished).

(2.2) THEOREM ([6]). A finite projective plane of order q2 admitting a

collineation group isomorphic to PSU(3,q) is Desarguesian.

(2.3) THEOREM ([9]). If the collineation group of a finite cyclic projective

plane contains more than one cyclic Singer cycle, then this plane is

Desarguesian.

3. PRELIMINARY RESULTS

(3.1) LEMMA. Let F = Fi(A,a) be a flag group. Then IFI < n.

PROOF. Let F act on (a) \ {A}. By the definition of a flag group, the

stabilizer of a point X in this set is I. Hence, IFI = IxFI < n.

In the rest of this section, we assume that G contains a flag group

F = F%(A,a) of order n.

(3.2) LEMMA. We have either n < 3 or G does not leave invariant any triangle

and one of the following holds.

120 Goncalves and Ho

(i)

(ii)

(iii)

(iv)

PROOF.

Fix(G) = ({A},{a}).

FIx(G) = (~,{a}) and G is a translation plane with respect to the line

a and G contains the translation group of E.

Dual statement of (ii).

Fix(#,~) and G is irreducible.

If G leaves invariant a triangle, then the action of F on the vertices

of this triangle shows that n < 3. Hence, we may assume that G does not leave

invariant any triangle.

Suppose Fix(g) ~ ({A},{a}) or (~,~). Then Fix(G) is either (~,{a}) or

({A},~). Assume Fix(G) = (~,{a}). Thus, there exists a conjugate of F not

fixing A. This implies that each point in (a) is a center of a flag group of

order n and axis a. Hence, G is transitive on the affine lines of the affine

plane (P \ (a), L \ {a}). By Wagner's theorem [7, p, 275], case (ii) holds.

A similar argument yields (iii) when Fix(G) = ({A},~).

(3.3) COROLLARY. If n is neither a translation plane nor a dual translation

plane, then G A = GA, a.

PROOF. This is clear from 3.2

(3.4) PROPOSITION. Each one of the following additional conditions implies

that G does not leave invariant any proper subplane.

(i) G contains a non-trivial perspectivlty.

(ii) IGI is even.

(iii) F = F%(A,a) is cyclic and Fix(G) # ({A},{a})o

PROOF. By way of contradiction, assume G leaves invariant a proper subplane

~o = (Po'Lo) of order m. As [A] \ {a} is an orbit of F, if A s Po" then

m + I ~ n, a contradiction. Thus A ~ Po" Hence [A I \ {a} does not contain

line of n o. A similar argument shows that a ~ L o and (a) does not contain any

any point of n o. Since [A] \ {a} is an orbit of F and F leaves ~e invariant~

we now get n = m 2 + m + 1. In particular, n is odd and not a square~ Thus~

any involution in G is a homology by [7, p. 95]. Let G be the eollineation

group of n ~ induced by G. By Roth's result [1, p. 171], we have G ~ ~ as

n = m 2 + m + 1.

Suppose (i) or (ii) holds. The perspectivities of G induce perspectivities on

n o. Since F acts transitively on Po' n o is Desarguesian and ~ contains the

little projective group ~ of n ~ by Wagner's theorem [7, p. 260]. Let r ~ L o

and let E < G such that E is the group of elations of n with axis r~ Hence, o


IEI = IEI = m 2. Any element in E is a product of 2 involutions in S. Thus

any element in E is a product of 2 involutiorial homologies in S. Therefore,

the fixed points of a non-trivial element in E belong to (r). This implies

that E acts semi-regularly on the n 2 points of F \ (r). Hence, m 2 divides

n 2. However, this contradicts the fact that n = m 2 + m + I.

Next, assume that F is cyclic, and Fix(G) # ({A},{a}). Thus, there exists a

conjugate H = F%(B,b) of F such that (B,b) ~ (A,a). Both F and H are cyclic

Singer groups of ~ . Since G ~ G and F * H, F * H. Hence, ~ is Desarguesian o o

and G is 2-transitive on Po by [9]. This implies that IGI is even, a case

which has just been treated. The proof of 3.4 is now complete.

(3.5) COROLLARY. If Fix(G) = (~,#) and one of the three conditions in the

statement of 3.4 holds, then G is strongly irreducible.

PROOF. This is clear from 3.4

(3.6) LEMM_~. Let W = {X s PIG contains a flag group F%(X,x) of order n for

some x} and A = {x c LIG contains a flag group Fi(X,x) for some X e P}. Then

W = A G and A = a G.

PROOF. By the definition of a flag group, we can find a prime p such that for

each X e W there exists an element g e G of order p with P(G) ~ W = {X}. From

[7, Lemma 13.5, p. 258] we get that W is an orbit of G. Hence W = A G. A G similar argument gives A = a .

(3.7) LEMMA. Assume Fix(G) = (~,#) and let H = F%(B,b) be a flag group of

order n such tbat A ~ b and B ~ a. Set J = (a) N (b). Then we have the

following.

(i) If J 6 A G, then P = A G and L = a G.

(ii) If J ~ A G, then IJGl ~ n(n + i) 2

PROOF. (i) Assume J = A g ~ A G. Let R = F g = F%(J,a g) Since (a) = {A} ~ jF

(a) ~ A G. By 3.6 (i) we have B c A G = jG. As (b) = {B} ~ jH,

(b) ~ jG = A G. As a g # a or b, we get [J] \ {a g} = a R or b R. Hence, all

~oints incident with a line in [J] \ {a g} belong to A G. Since (a) < A G,

(a g) ~ A G. Therefore P = A G as required. A similar argument yields L = a G.

(ii) Since jG > (ag) \ {Ag } for all g ~ G, we have

iJGl > n + ... + i n(n + i) 2

(3.8) COROLLARY. Assume G contains a perspectivity in addition to the

conditions in 3.7. If J ~ A G, then ~ is Desarguesian and G contains the


little projective group of ~.

PROOF. This is an immediate consequence of 3.7 (i) and Wagner's theorem [7,

p. 260]

(3.9) LEMMAo If A is the center of a perspectivity in G, then G contains an

elation.

PROOF. This comes from the dual statement of [7, Corollary I, p. 104]o

(3.10) LEMMA. Let I # t e G be an element of prime order p with p~no Then

the following hold.

(i) If CF(t) # I, then t is planar and the order of Fix(t) is bigger than

or equal to ICF(t)I

(ii) t % CG(F) and ICF(t)I < ~'n.

(iii) If ICF(t) I = ~'n-, then t is a Baer collineation.

(iv) If t is a Baer collineation in NG(F), then ICF(t) I =

PROOF. (i) If CF(t) ~ I, then ({A},{a}) < Fix(t). Since p~n, t fixes some

point in (a) \ {A} and some point X outside a. As CF(t) is a flag group with

axis, a X CF(t) does not belong to a line. Since CF(t ) acts on P(t), Fix(t) is

a subplane. The action of CF(t ) on [A] ~ L(t) implies that the order of

Fix(t) is bigger than or equal to ICF(t) I as desired.

(ii) If t 6 CG(F), then (i) implies that the order of the subplane Fix(t) is

bigger than or equal to IFI = n, a contradiction. If CF(t) = I, then clearly

ICF(t)I < ~ If CF(t ) # I, then (i) implies ICF(t)I < ~-n by [7, Theorem

3.7, p. 81]. Therefore, ICF(t)I < q'~as required.

(iii) If ICF(t)l = ~ then (i) implies that the order m of Fix(t) satisfies

m > ICF(t)I = ~-~n. By [7, Theorem 3.7, p. 81] again we get that m = AF-nn, and

Fix(t) is a Baer subplane.

(iv) Since t 6 BG(F), ({A},{a}) < Fix(t). Let I ~ j 6 F such that for some

X 6 P(t) \ {A} we have X k 6 P(t) \ {A}. Then X k e P(t) ~P(t k)

= p(t -I) ~ P(t k) < P([t,k]). As [t,k] c F, P([t,k]) = {A} if It,k] * io By

the choice of X we must have [t,k] = I. Now F is sharply transitive on

(a) ~{A}. Let Y 6 Q = (a) ~ P(t) ~k{A}. For any Z E Q, there is f ~ F such

that Z = Yf. hence, f E CF(t ) by the above argument. This shows that Q is an

orbit for CF(t) and so IQ I = ICF(t) I. Since IQ I = order of Fix(t), !CF(t) I =

IQ I = ~-nn as desired.


4. PROOF OF THE THEOREM

In this section we assume that G contains a nontrivial perspectivity, and a

flag subgroup F = F%(A,a) of order no Assume further that Fix(G) = (~,~).

Under these conditions we prove that ~ is Desarguesian.By 3.5, G is strongly

irreducible. Let M be the unique minimal normal subgroup of G provided by

2.1. By way of contradiction, assume that ~ is not Desarguesian. By 2.1 and

the classification of finite simple groups, we get that M is isomorphic to one

of the following groups: Z 3 • Z3; Ar, r e {5,6,7}; PSU(3,q); PSL(2,q); J2"

Step i. M ~ Z 3 x Z3.

PROOF. Since G/M is isomorphic to a subgroup of GL(2,3), IGI divides 24.33 .

As M is abelian and Fix(~,~), we have F ~M = I. This implies that

n = IFI ~ 3. Thus ~ is Desarguesian, a contradiction.

Step 2. M ~ Ar,r ~ {5,6,7}.

IAut(Ar)/Arl is a power of 2 and n is odd, F ~ M. If r = 5, PROOF. Since

then n E {3,5}, a contradiction. If r = 6, then n = 9. However, N

M A 6 PSL(2,9) and n = 9 imply that ~ is Desarguesian by [I, p. 186], a

contradiction. Assume r = 7. Since projective planes of order less than or

equal to 7 are Desarguesian, n > 7. By Bruck-Ryser's theorem [7, p. 80],

n ~ 21. Since n is odd, n = 9 in this case. Hence, F SyI3(M ). There

exists an involution t in NG(F ) inverting F. If t is a Baer involution, then

3.10 (iv) implies that ICF(t)I = 3, which is impossible. Thus, t is a

homology. Hence, all involutions in M are homologies. There exists a 3-

element in F which commutes with an involution. Since no flag collineation

can commute with a homology, we get a contradiction.

Step 3. M ~ PSU(3,q).

PROOF. If q + {3,4}, then ~ or the dual of ~ contains a G-invariant unital

U = (Q,B) of order q by Proposition 5.3 of [5]. Thus IQI = q3 + 1 and for

each P ~ Q, I[p] ~ BI = q2. First assume that q > 5. Since ~ is Desarguesian

if and only if its dual is Desarguesian~ we may assume withut loss of

generality that a G-invariant unital U = (Q,B) of order q is contained in ~.

This implies n > q2 as [[P] N B I = q2. Suppose A 6 Q. Since F acts semi-

regularly on B ~ [A] \ {a} and n > q2 = I[A] ~ BI ' n = q2. Therefore, ~ is

Desarguesian in this case by 2.2. Hence, we may assume A + Q. If a c B, then


(a) ~ Q is F-invariant. Since A ~ Q, (a) ~ Q ~ {A}. This implies

q + 1 = I(a) - QI > IF > q2, which is impossible. Therefore, F acts semi-

regularly on the q3 + 1 points of Q and on the q2(q2 _ q + I) lines of Bo 2

This gives n = IFI divides q2 _ q + I~ which contradicts n > q .

Assume now q = 3. Proposition 5.3 of [5] implies that the non-trivial

perspectivities of ~ in G are precisely the involutory homologies of M such

that distinct involutions have distinct centers and axes. By 2.1 (ii) and

3.9~ we see that F acts semi-regularly on the set of q2(q2 . q + I) = 9~7

involutory homological centers of M in ~. By 2.1 (v), we have (3,n) = I.

Hence nit and ~ is Desarguesian, a contradiction. i

Finally, let q = 4. Then IMI = 26.3.52.13. Let P s SyI2(M ). Proposition 5.3

of [4] implies that distinct involutions in M have distinct centers and axes,

Z(P) ~ Z 2 • Z2, and there is a planar element ~ of order 5 in CG(Z(P)).

Since flag collineations cannot commute with any homology, F acts semi ~

regularly on the set of 3.5.13 involutions of M. Hence, n divides 3.5.13.

Since F < Aut(M) and n is odd, [F:F ~M] = 1 or 3. If IFI = 5.132 then

F < M. However, the normalizer of Sylow 13-subgroup has index 43.52 in M,

which implies that M does not contain any subgroup of order 5.13~ Hence,

n c {3,5,13,15,39}. As ~ is not Desarguesian, n s {13,15,39}. Since

commutes with an involutory homology Fix(m) has odd order m. Also by [I, p.

171], we get that n # 13 or 15, and m < 4 for n = 39. Hence, m = 3. However,

5~39 - 3, a contradiction. This completes the proof of Step 3.

Step 4. M PSL(2,q).

PROOF. By [5] we may assume that q is an odd prime power, and if

q % {7,9}, then involutions of M are homologies such that distinct involutions

have distinct centers and axes.

Since PSL(2,9) ~ A 6 which has been treated in 4.2, we may assume q # 9. If

q = 7, then F < M as IG/M I is a power of 2. Hence, n = IFII 21. Since n # 21

by Bruck-Ryser's Theorem [7, p. 80], n 6 {3,7}. Therefore, ~ is Desarguesian~

a contradiction. Hence, we may assume q ~ 7.

Thus, F acts semi-regularly on the set I of involutions in M. Since IF 1 is

odd, F < M �9 K, where M ~ K = i and K ~ Aut(GF(q)). Let q = pS. Then K is a

cyclic group of order s. Assume F ~M = I. Then n = IFI < s, and

2 n + n + 1 < s 2 + s + I. Since p is an odd prime~ pS (pS

o

I11 Hence, P = I and ~ is a Desar~esla~ by s2+s+l<- 2


Wagner's theorem [7, p. 260]. Therefore, we may assume F ~ M # I. By 2.1

(ii), we may assume that G does not contain elations. Hence, A is not the

center of a perspectivity by 3.9. Let I * ~ c F ~ M with prime order p.

Since Fix(G) = (~,~) and F acts semi-regularly on A G\{A}, IA G\{A} I = i + kn

for some positive integer n.

Suppose r = p. Let ! * ~ E S ~ Sylr(M). Then S is an elementary abelian p-

group, and Fix(S) = ({A},{a}). Let N = NM(S). Since N is a Frobenius group,

= I~NI. Assume ~ i s ~ even. Let ~ be an involution in N. Then ~ is [N:S]

a homology. Denote the center of ~ by C. Thus, C ~ A G. Since ~ 6 N,

C ~ (a) \ {A}~ Hence, n(n + i) < IcGI = IcMI = q(q + i) by 3.7 (ii). This 2 2

implies that n < q. There is a dihedral subgroup of order q + i of M, which

is a maximal subgroup. Since involutions are homologies, this subgroup fixes

at least one point. Let B be a fixed point of this subgroup. Hence,

IBM[ _ q(q - I). P As Fix(M) = (~,~) and r = [MAI , B ~A G. Hence, I l 2

n 2 q(q i) q(q + i)

2 + 2 i i Q i i i

= i + kn + q2. Since q > n, the last equation implies that q = n. Therefore,

is Desarguesian by [i, p. 186]. Hence, we may assume that

q(q - I) is odd. Thus, [~N I q - I N ( -I)N 2 = -- ~ , and S = {I} ~ ~ ~ is a flag

subgroup of order q. By 3.1, q ~ n. Let Sylp(M) = {S ~ = S,SI,...,Sq}. If an

element f ~ 1 in F normalizes some S i with i > 0, then Fix(Si) = (Fix(So)) f

= ({A},{a}) f = ({A},{a}). As M = <So,Si> ,

contradicts Fix(M) = (~,~). Hence, F acts

so IFI ~ q. Thus, q = n in this case, and

this implies that M fixes A which

semi-regularly on Sylp(M) \{S}, and

is Desarguesian by [I, p. 86], a

contradiction. Therefore, we may assume r # p. Let T c SyI2(M ) and let C be

the center of an involutory homology in Z(T). Then C ~ A G, and M C is a q+e

dihedral subgroup of order q + g, where s = E1 such that ---~-- is odd. Let L

be a dihedral subgroup of M of order q - g. Since distinct involutions have

distinct centers and axes, the axes of L pass through a common point

B. As L is a maximal subgroup in M and Fix(M) = (~,~), [B] contains exactly q -

2 axes of involutory homologies of M. If B = A G, then the action of F g on

the axes of involutory axes in [B] implies n q - g 2 Thus, q = 2n + ~.

= - ~) = q2 2 n 2 However, IAGI + [cGI q(~_L) + q(q 2 = (2n + ~) > + n + i, a

contradiction. Therefore, B ~ A G, and so 2

n + n + I > IAGI + IBGI + IcGI = (kn + I) + This implies that q < n. I I I I I I q2

Since r # p and dihedral subgroups of the same order q ~ ~ are conjugated in

M, we may assume ~ c M C or L. By 3.7 (ii), we have either IBGI > n(n + I) 2 or


!cGI ~ n(n + I) 2 Both of these imply n < q. Therefore, n = q and E is ; i

Desarguesian by [I, p. 186], a contradiction. This completes the proof of

Step 4.

Step 5. M J2"

PROOF. Proposition 2 of [Ii] implies that a central involution ~ of type (21)

of M satisfies [~M I = 32.5.7. Moreover, these involutions are homologies such

that no two share a common center or a common axis. Also, the same

proposition implies that involutions of type (22) are planar. In particular,

n is a square. Since a flag collineation doesn't commute with any homology~

the action of F on a M yields ni32.5.7. As n is a square, n = 9. By

Proposition 2 of [Ii] there exists a generalized perspectivlty w in M or order

5 such that iP(w) I ~ 3 (mod 5). However, this contradicts n = 9o This

completes the proof of Step 5.

The proof of our theorem is now completed by the Steps I to 5.

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[13] Wagner, A. On perspectivities of finite projective planes, Math. Z. 71 (1959), 113-123.

[14] Wagner, A. On affine line transitive planes, Math. Z. 85 (1965)~ 451- 453.

Adilson Goncalves Depto. de Matematica Univ. Fed. de Pernambuco Recife, PE Brasil

Chat Yin No Dept. of Mathematics Univ. of Florida 201 Walker Hall Gainesville, Florida 32611 U.S.A. (and Univ. of Toronto)

(Eingegangen am 6. Mai 1985)

on flag collineations of finite projective planes

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