Journal of Geometry Vol. 11/2 1978 Birkh~user Verlag Basel

ON CENTRAL COLLINEATIONS OF DERIVED SEMIFIELD PLANES

Norman Johnson

Let ~ denote the collineation group generated by the set of all affine central col!ineations in a derived semifield plane. We present a characterization of the Hall planes in terms of the order of ~. This essentially allows the exten- sion of the theorems of Kirkpatrick and Rahilly on generalized Hall planes to arbitrary derived semifield planes. That is, a derived semifield plane of order qS is a Hall plane pre- cisely when it admits q+l involutory central collineations.

I. Introduction

Our main result in this article is:

(2.5) THEOREM: Let w be a derived semifield plane of order

q2 q/4,8 or a Mersenne prime. Let ~ denote the colline-

ation group o_~f ~ which is generated by the sat . o_~f all affine

central Collineations. Then I~I contains a prime p-primi-

tive divisor of q2-1 i_~f and qnlz i_~f ~ is a Hall plane.

In El3], Kirkpatrick defines a class of translation

planes called generalized Hall planes. These planes are of

order q2 and admit a collineation group ~ which fixes a

Baer subplane ~O pointwise and acts regularly on the infi-

nite points of the line at infinity which are not in the

associated Baer subplane (i.e., on ~-wON ~ ). In a later

paper [12], he shows that a generalized Hall plane w of 2

odd order q is a Hall plane precisely when ~ admits

q+l affine involutory homologies with centers on wO G ~ .

Furthermore, Rahilly [16] has shown that the result also

holds for generalized Hall planes of even order (i.e., by

replacing "involutory homologies" by "elations").

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2 JOHNSON

The Hall planes are well known to be exactly the class

of translation planes which may be "derived" from the Des-

arguesian planes. That is, constructed from the Desar-

guesian planes by the replacement of one derivable net.

In a similar way, it is also known that finite gener-

alized Hall planes are derivable and may be derived from

semifield planes whose coordinate semifields are of dimen-

sion i or 2 over their middle nuclei. (See Jha [7],

Johnson [8], Rahilly [16].)

The results of Kirkpatrick and Rahilly may thus be

stated as follows:

Let ~ be a semifield plane of order q2 which con-

tains a derivable net ~ which is coordinatized by a sub-

field of the middle nucleus of the corresponding semifield.

Let w be the derived semifield plane obtained by replac-

ing ~ by its unique replacement net ~. Then w is a

Hall plane if and only if w admits q+l affine involu-

tions whose axes are the q+l components of ~.

Thus, this suggests that a stronger characterization

of the Hall planes may be obtained by considering affine

central collineations of arbitrary derived semifield planes.

Johnson and Rahilly [Ii] have studied this problem for

planes of even order and have shown an arbitrary derived

semifield plane w of even order q~ ~16 is a Hall plane

if and only if w admits q+l distinct affine elations.

Using (2.5), we obtain the corresponding extension of

Kirkpatrick's theorem for planes of odd order except that

our argument does not apply to planes of characteristic

three. That is, let w be a derived semifield plane of

order q2 q~4 or 3 r. Then w is a Hall plane if and

only if w admits q+l distinct affine involutory cen-

tral collineations.

For the remainder of this article, let w be a semi-

field plane of order q2 and assume ~ is a derivable net

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JOHNSON 3

of ~ which contains a shear axis of ~. Let w denote

the translation plane obtained by replacing ~ by its

unique replacement net 7. In this case, w is said to

be a derived semifield plane.

2. Central Collineations of Derived Semifield Planes

(2 .1) . For (2.1) and (2 .2 ) , see [ i i ] , (2.1) and (2 .2 ) . Assume ~ admits an affine central collineation o. If

q ~4 and ~ is 'nonDesarguesian then the axis and center

of ~ is in 7.

(2.2). If w admits affine elations then q is even.

2r 2 (2.3) PROPOSITION. Let w have order p =q ~16 and

assume w is not a Hall plane. Let ~ denote the collin-

eation group which is ~enerated by the affine central col-

lineations of w. Assume there is an element • 6~ such

that I• i_ss a prime-7~q -I but I• L.C.M.(pS-!), sI2r,

s <2r. (That . i_ss, IXl is a p-primitive divisor of q2-1.)

Then there i_~s a D esarguesian plane ~ such that the compo-

nents of ~ which are fixed by X are precisely those of

EOw. Moreover, the degree of EDw > l+q.

Proof. Every central collineation �9 in ~ must fix

7 and thus must act as a collineation of ~ which fixes

7 O~ pointwise (i.e., T is a Baer collineation in W).

Thus, ~ fixes at least l+q ~3 components of ~ and by

[15] corollary to Theorem l, the element X (which thus

also fixes ~ 3 components) must also be a collineation of

a Desarguesian plane ~ such that the components of

which are fixed by X are exactly those of ~ N~.

Let J denote the shear subgroup of W which fixes

~. X fixes ~ D~ pointwise so if g 6J and maps P

onto Q where P,Q E 7 D ~ then gX also maps P onto

Q. But, there is a unique elation which maps P onto Q

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4 JOHNSON

and has a fixed center. Thus, gX =g so M commutes

with any elements of J and IJ] = q. So, X permutes

the J-orbits on ~ of ~ and fixes the J-orbit

N~-(~) (where we denote the shear center by (~)).

Since there are q-i remaining J-orbits (J is semireg-

ular on ~- (~)) and ]x]~q-1, X must fix some other

orbit F. Since X centralizes J~ J must also permute

the points fixed by X. IF1 =q so X must fix some

point of F (note we may assume iXl is a prime) and

hence must fix every point of F. Thus~ X fixes ~ U F

pointwise.

(2.4) PROPOSITION. Under the assumptions o_~f (2.3), there

is a shear subgroup ~ o~f ~ which is resular on

~ n ~ - {(~)}.

Proof. Recall that ZDw is exactly the set of com-

ponents which are fixed by X. Each J orbit (in the

notation of (2.3)) which is fixed by M is fixed pointwise

by M and conversely, if X fixes a point of some J-

orbit on ~ it must fix that J-orbit. Thus, Z D w con-

sists of (~) U a set of J-orbits. Let g be some shear

that maps a point P in ZDwD~ onto a point Q in

ENw D~ . Since X fixes P,Q and (~) it follows that

X must commute with g and thus that g must fix the set

of points on ~ of W that are fixed by X- That is, g m

must fix Z 0 ~ and clearly there is a regular elation or

shear subgroup J as maintained.

(2.5) THEOREM. (See statement in Section i.)

I - ~pS-1 for s 12r Proof. Note that if IX] p2r 1 and JiM I pt-1 then then IXl~pt-i for any t <2r (i.e., if

IXIl(p2r-l-l,pt-l) :p(2r't)-l). In the notation of (2.A),

let 17] =pr+m where q=pr and i <m~r. There are

p(r-m)-i T-orbits on ~ which are permuted by •

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JOHNSON

different from ~AT- [(~)]. If X fixes any of these

orbits it will fix the orbit pointwise. Thus,

IXllp(r-m)-i implies r-m =i so that i~ [ =p2r =q2. Hence the degree of ENw- [(~)} is q2 and thus ZD

= E, ~ is Desarguesian, and ~ is Hall.

(2.6) COROLLARY. (See Johnson and Rahilly [ii] (2.14).)

A derived semifield plane ~ of even order q2 ~16 o_.~r 64

is a Hall ~lane if and only if n admits q+l affine

elations.

Proof. By (2.2), q is even. By (2.1) and Gleason's

theorem [5], the order of the group ~ generated by the

elations is divisible by q+l. By Birkhoff and Vandiver

[1], if q ~8, q2-1 always has a prime 2-prlmitive

divisor which must divide q+l. Thus, (2.6) follows di-

rectly from (2.5).

Note that (2.6) is also true for q2 =16 or 64 if

the elations are assumed to have centers in the associated

derivable net. However, the proof for q =8 requires more

detailed analysis of the group ~ (see Ill] (2.14)).

With regard to homologies not of order two we note

the following:

(2.7) COROLLARY. Let w be a derived semifield plane of

order q2 q ~4 o_.~r 8. Then ~ is a Hall plane if and

onl[ if w admits at least q+l affine homologies of

order k >2.

Proof. By (2.1), either w is Hall or there exist

exactly q+l homologies of order k >2 with centers and

cocenters in ~. By 0strom [14], it follows that we have

an orbit on ~ of length q+l/2 or q+l. Now since

q ~8, either q2-1 contains a p-primitive divisor or q

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6 JOHNSON

is prime. Thus, by (2.5), we have the result.

Note that (2.7) is also valid for q=8 since the

above assumptions impl~ that w admits elations with q+l

centers. Note also that one of the derived semifield

planes of order 16 admits an affine homology of order 3

(see, e.g., [lO]).

We now briefly book at the group structure of the

group generated by the affine homologies.

(,~.8). PROPOSITION. Let ~ be a derived semifield plane

of order p2r = q2 p odd, which i_~s not ~ Hall plane. Let

denote the group generated by the affine central collin-

eations of ~. Then ~ is isomorphic . to a subgroup of

GL(2,q) and. ~I~ O~ (~ the corresponding derivable net)

i.~s isomorphic to a subgroup of PIGL(2,q) where

p~I~l~ n~ I. Thus, ~I~ AS ~ a dihedral group of order

2(q#l) or is isomorphic t_~o A4, S~ or A 5.

Proof. The net ~ is Desarguesian by Foulser [3].

Let wO be any Baer subspace of ~. Every affine central

collineation of w fixes ~ and thus is a collineation of

w O (see (2.1)). Thus, �9 fixes n o and induces a col-

lineation group on ~O" Moreover, since ~ fixes each

such Baer subplane of ~, ~lw O is faithful on n O. So

~GL(2,q) (~ is generated by elements of GL(2,q)). Let

be ~I~ N~ . If pll~l then ~ contains an element i

of order p which acts as an elation on each Baer subplane

of ~. (Note that the following may be verified: if ~6

and IXI is prime and relatively prime to q-i then there

is X i 6~ such that Ixil = Ixil (the image in $) = IXI

and X i is a preimage of ~i.) If o is a Baer p-ele-

ment whose fixed points are on components of ~ then its

Baer axis is also in ~. Since we can assume I~ 1 =Iol =p,

we have a contradiction. So ~ must be an elation of

which is contrary to (2.2). The structure of ~I~ N~ now

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JOHNSON y

follows from Dickson (see [6]).

(2.9) PROPOSITIO N . Let ~ be a derived semifield plane

of odd order p2r Le t x be an element ~enerated by

affine central collineations such that IXIlp-l. Then if I

IX1 i_~s ~rime ' or more generally X leaves a component of I

the derived net ~ invariant then either X is a kern

homology o~r IXI =2.

Proof. Let • fix the component (x=0) of ~. X

must fix each Baer subplane of ~. If X fixes (x=0)

pointwise then there must be another component of ~ upon

which X is fixed point free. Let ~ replace ~. Then

X fixes ~ N ~= pointwise and acts like a kern homology on

some Baer subplane wO of 7.

The semifield plane ~ admits the (O,~)-homologies

represented by (x,y) r for all a 6GF(p) and

aa must leave every Baer subp!ane of ~ (incident with

O) invariant.

Thus~ we may assume that X 6 (~a>I~ O. And there is a

kern homology of ~ such that aX fixes an affine point.

Since both ~ and X fix ~D~ pointwise, it must be

that ~X = i or ~X is a Baer collineation of order

LCM(I~I,IXI) (~X=Y~ since ~ is a kern homology of ~).

So~ either X is a kern homology of ~ which fixes

all Baer subplanes of ~ and is thus a kern homology of n

or ~X fixes a Baer subplane wO of ~ pointwise, n O

is Desarguesian and coordinatizing ~ so that n O is

coordinatized by F ~ GF(q), q =pr, we have that cX must

induce an automorphism of the coordinatizing semifield J

which fixes F pointwise. Since the net ~ is Desar-

guesian, J is a right 2-dimensional vector space over F.

Since I GI and IXIlp-i then l~Xllp-l. By choosing

n O to be (x=O) and the cocenter of oX to be (0) in

w it is direct that the automorphism induced by GX~ 8~X ~

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8 JOHNSON

may be represented by t--~t~ where ~ E F, ~Ip-1 and

{1,t} is a basis for J over F. Let t 2 =tf+g, f,g 6F.

Then applying the automorphism 0oX we see that (t~)(t~)

= (tB)f+g. Since J is a right vector space over F, we

have (tS)f+g=t(~f)+g and since ~Ip-1, ~ is in GF(p)

so that (tS)(t~) =t282 =(tf+g)82 =t(f82)+g82. Equating

vector parts, we have: f~2 =~f and g82 =g. But, g~O

so 2=1. so I I:2:leo•215215 Thus, ]XI =2 and we have the proof to (2.9).

Let w be a derived semifield plane of odd order 2 pZr q = that is not Hall and admits q+l involutory

homologies. Let ~ denote the group generated and

~ = ~I~ A~ . By (2.8), ~ is, either A~,S4,A 5 or is

dihedral of order 2t where tlq~ i.

There are at least q+i/2 distinct involutions in

~ . So if ~ is dihedral, t or t+l ~q+i/2 depending

on whether t is odd or even. Let tk=q el and assume

k ~ 3. (If t is odd then clearly k - I or 2 and q �9 1

= q+l.) If t is even and k ~3 then (q• =(q+l) and

it follows that q ~5. Thus, q=3 or 5 and since semi-

fields of prime square order are fields it is direct that

is a Hall plane in this case.

If t =q+l or q+i/2, ~ is Hall by (2.5). Thus,

t = q-I or q-i/2.

Thus, suppose on ~ ~ ~ we obtain a dihedral group

~, of order 2t where t =(q-l) or (q-I/2). The cyclic

group of order t fixes two points, say (~) and (0),

that are interchanged by the normalizer. Let ~ denote

the ((~),y=O)-involutory homology. Since ~ is a Desar-

guesian net, ~ is (x,y)-~(x,y(-l)). The ((O),x=O)-

homology is (x,y)--~(x(-l),y) (i.e., there must be a

((O),x=O)-involutory homology). Since a must fix all

homology pairs, the pairs may be represented by ((~),(-~))

146

JOHNSON 9

if a~O and the involutions by (x,y) o~ ~(y _i x= )

(recall the involutions fix the pair [(~),(0)]).

Consider caa8 : (x,y)--*(x(aB-l)-l,y(a8-1)). ~=o B is

obviously not a kern homology if ~%8. Clearly, l~l~l

= 181. Let 8 be an element of odd prime order Ip-l"

Then ~l~ is not a kern homology and we have a contradic-

tion by (2.9).

The only remaining case is when p-i =2 r for some r.

But, in this case ~=~^ fixes a component of ~. We may

choose ~ so that l~I =p-!. Thus, if the characteristic

3~ this particular situation cannot occur.

Assume ~ is A4, S 4 or A 5. There are at least

q+i/2 involutions in ~ but in A 4 there are exactly

three. Thus, if ~ is A t then q+i/2 < 3 so that

q = 3 or 5. Similarly, q+i/2 < 9 if ~ is S t and

q+l/2 ~15 if ~ is A 5.

Given an involution a i of ~ let the number of

preimage involutions in ~ ~e k i. We know that k i =i

or 2 for each i. Furthermore, if ci has two distinct

preimage involutions then so does any conjugate. There are

exactly q+l affine involutions that generate ~. So if I

l is the number of involutions in ~ then ~ =q+l. i=l

In S 4 the involutions are in two conjugacy classes

of lengths 3 and 6. Since q+I/2 ~9, q ~iZ and if

is not Hall then q is odd and not prime. So q =9. o

Thus i=~iki = I0 which implies ki = 2 for exactly one

value i~ which is contrary to the above.

In A 5 the involutions are all conjugate, so in this 15

case q+l = ~ k. so that q+l =15 or 30. Hence q=29 i=l l

and w must be Hall in this case.

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i0 JOHNSON

(2.10) THEOREM. Let w be a derived semifield plane of

order q2 and characteristic > 3. Then w is a Hall

plane if and only if w admits q+l distinct affine

invo!utor ~ homologies.

REFERENCES

[io]

[ii]

[12]

[13]

[14]

[15]

[i] Birkhoff, G.D. and Vandiver, H.S.: On the integral di- visors of an-b n. Ann. Math. 5(1904), 173-180.

[2] Foulser, D.A.: Baer p-elements in translation planes. J. Algebra 31(1974), 354-366.

[3] Foulser, D.A.: Subplanes of partial spreads in trans- lation planes. Bull. London Math. Soc. 4(1972), 1-7.

[4] Ganley, M.J.: Baer involutions in semifield planes of even order. Geometriae Ded. 2(1974), 499-508.

[5] Gleason, A.M.: Finite Fano planes. Amer. J. Math. 78 (1956), 797-807.

[6] Huppert, B.: Endliche Gruppen I, Berlln-Heidelberg- New York, Springer 1967.

[7] Jha, V.: On the automorphism groups of quasifie!ds. Thesis, Westfield College, Univ. London, 1972.

[8] Johnson, N.L.: A characterization of generalized Hall planes. Bull. Austral. Math. Soc. 6(1972), 61-67.

[9] Johnson, N.L.: Collineation groups of derived semi- field planes, I, II, III, and correction. Arch. Math. 24(1973), 429-433; 25(1974), 400-404; 26(1975), i01- 106; (to appear).

Johnson, N.L.: Translation planes constructed from semifield planes. Pacific J. Math. 6(1971), 701-711.

Johnson, N.L. and Rahilly, A.: On elations of derived semifield planes. Proc. London Math. Soc. to appear).

Kirkpatrick, P.B.: A charactrrization of the Hall planes of odd order. Bull. Austral. Math. Soc. 6 (19~2), 407-415.

Kirkpatrick, P.B.: Generalization of Hall planes of odd order. Bull. Austral. Math. Soc. 4(1971), 205- 2o9.

Ostrom, T. G.; Finite translation planes. Lecture Notes in Mathematics, Springer-Verlag, #158, 1970.

Ostrom, T.G.: Linear transformations and collineations of translation planes. J. Algebra (3) 14(1970), 405- 416.

148

[ 16]

JOHNSON Ii

Rahilly, A.: Finite generalized Hall planes and their collineation groups. Thesis, Univ. of Sydney, 1973.

Norman Johnson Department of Mathematics The University of Iowa lowa City, lowa 52242

(Eingegangen am 12. Dezember 1976)

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