•
,UNIVERSITY OF NORTH CAROLINA
Department of Statistics
Chapel Hill" N. C.
THE CONSTRUCTION OF TRANSLATION PLANES
FROM PROJECTIVE SPACES
by
R. H. Bruck, University of Wisconsin
and
R. C. Bose, University of North Carolina
Novem~er 1963
National Science Foundation Grant No. GP 16-60 and AF-AFOSR-Grant Noo 84-63
Can eve.ry (non-Desarguest. an) projee.,tJVe p~' ~ imbedded
(in some natural, g~anet.ric ff.hion) in af1J8sarguesian)
projective space? The Ite,titn is ne..WI~.ut . ,trnp
. · ortant, for,
if the answer is yes, tfo entirely SEijifia'Wt fields of research can be uni ted~ 1his paper ;p~1des. conceptually
simple geometric construction 1vh1dti',Y1elds an affirmative
answer for a broad class of pla.uer. A plane 1t is given
by the construction precisely when 1t is a translation
p1ar1e with a coordinatizing :right Veblen-Wedderburn system
which is finite-dimensional over its left-operator skewfield. The condition is satisfied by all known translation
planes, including all finite translation p1aneso
This research was supported by the National Science Foundation Grant No.
GP 16-60 and the Air Force Office of Scientific Research Grant No. 84-63.
Institute of Statistics
Mimeo Series No. 378
•
THE CONSTRUCTION OF TRANSLATION PLANES
FROM PROJECTIVE SPACES
by
R. H. Bruck, University of Wisconsin
l
and
1
R. C. Bose, University of North Carolina
--------~--------------------------------.---------------------------------------
1.
Introduction.
One might say, with some justice, that projective geometry,
in so far as present day research is concerned, has split into two qUite separate
fields.
On the one hand, the researcher into the foundations of geometry tends
to regard Desarguesian spaces as completely known.
Since the only possible non-
Desarguesian spaces are planes, his attention is restricted to the theory of
projective planes, especially the non-Desarguesian planes.
~
On the other hand;stand
all those researchers - and especially, the algebraic geometers - who are unwilling
to be bound to two-dimensional space and uninterested in permitting non-Desarguesian planes to assume an exceptional role in their theorems.
For the latter
group of researchers, there are no projective spaces except the Desarguesian
spaces.
In the present paper we present a construction Which, we hope, may do just a
little to span the chasm between the two fields of projective geometry.
Speci-
fically, we show how to construct a class of non-Desarguesian planes (which occur
most naturally in affine form) in terms of the elements (certain points and certain projective subspaces) of (Desarguesian) projective spaces of even dimension.
The construction is given in Section
4.
~eceived by the editors October
, 1963. This paper was written while
Bruck ~s spending a yearts leave with Bose at the University of North Carolina.
. The authors had the support of the National Science Foundation Grant No. GIl 16-60
and the Air Force Office of Scientific Research Grant No. 84~63.0
•
2
The researcher in the theory of planes will want to know what planes we have
constructed.
There are two answers:
(1)
Only translation planes.
(2) All
finite translation planes. and, more specifically, precisely those translation
planes coordinatized by a right Veblen-Wedderburn system which is finitedimensional over its left-operator skew-field.
See Sections 6, 7; in particular,
Theorem 7.1.
Since this paper has two authors, the following remarks may be appropriate:
The present construction hinges upon the concept of a spread (of an odd-dimensional
projective space; see Section 3.) This concept was introduced by Bruck, with the
construction of planes in mind (and as a natural sequel to the concluding part of
his 1963 lectures to the Saskatoon Seminar of the Canadian Mathematical Congress;
see Bruck [lJ) but with a different objectivec
The construction presented here
is entirely due to Bose and evolved, essentially, from considering spreads of
3-dimensional projective space (which are maximal sets of skew lines) in terms of
5-dimensional project.ive space.
And, finally, the analytical details of the paper,
including the precise relation with translation planes, were supplied by BruCk.
The first example of a non-Desarguesian translation plane was given in 1907
by Veblen and Wedderburn [2J.
The first examples of spreads (in the sense used
in this paper) seem to have been given in 1945, 1946 by C. Radhakrishna Rao [3a,
3bJ.
(Equivalent examples are given in Section 3.
However, in [3bJ, Rao also
gives a solution of the original Kirkman Schoolgirl Problem by partitioning the
35 lines of projective 3-space over GF(2)
into 7 disjoint spreads, each
spread consisting of 5 skew lines containing, by threes, the 15 points of space.)
The first published use of spreads for the construc'tion of non-Desarguesian planes
is that in the present paper.
•
3
2.
Projective spaces in terms
~vector
spaces.
Since most algebraists
are more familiar with vector spaces than with projective spaces , we wish to
recall a classical representation.
This representation is thoroughly studied, for
example 1 in Baer [4J.
Let F be a skew-field (that is , an associative division ring which mayor
may not be commutative) and let V be a vector space with F as a ring of (say)
left-operators.
The dimension of V over F may be finite or infinite but
(to avoid trivialities) should be at least 3.
space E
= E(V/F)
in the following manner:
From V we define a projective
A point (or a.dimensional projective
subspace) of E is a I-dimensional vector subspace of V over F. More generally,
for each non-negative integer
s, an s.dimensional projective subspace of E is
an (s+l)-dimensional vector subspace of V over F. And incidence in E is defined in terms of the containing relation in V.
theorem of E.
The axiom of Desargues is a
The axiom of Pappus is valid in E precisely when F is a field.
Conversely, if d > 2 is a positive integer and if E is ad-dimensional
projective space (and if E satisfies the axiom of Desargues in case
d=2) then
there exists a skew-field F, uniquely defined to within an isomorphism, and a
(d+l)-dimensional vector space V over F such that E is isomorphic to
E(V/F).
Since, by the Theorem of Wedderburn, the only finite skew-fields are the
Galois fields
GF(q), one for each prime-power
q, the only finite d-dimensional
Desarguesian projective spaces are the projective spaces PG(d, q), one for
each d ~ 2 and each prime-power
V (d+I)-dimensional over GF( q) •
q, where PG(d,q) is defined as above, with
•
4
3. Spreads. Let
dimension
2t-l.
sapces of
~.
~
be a (finite or infinite) projective space of odd
Let S be a collection of
We call S a spread of
tained in one and only one member of
tive line and L
~
~.
(t-l)-dimensional projective sub-
provided that each point of
- Note that, if t=l,
~
~
is con-
is a projec-
bas precisely one spread, namely the collection consisting
of all the points of
~.
In the special case that L = PG(2t-l,q), a simple calculation shows that
a spread of ~
is merely a collection of 1 + qt
distinct (t-l)-dimensional
projective subspaces which are skew in the sense that no two have a common point.
The existence of a spread of PG(2t-l),q) may be shown qUite simply in
terms of the representation described in Section~. Set
K = GF(qt)
L = GF( It).
Let
be the unique subfield of L of indicated order and let F = GF(q)
be the unique subfield of Land K of indicated order.
Then L is a 2-
dimensional vector space over K, and K is a t-dimensional vector space over F,
and L is a (2t)-dimensional vector space over F.
isomorphism,
PG(2t-l, q) = ~(L/F)
Hence, in the sense of
and PG(l, qt) = ~(L/K).
The set,
S, of all
l-dimensional vector subspaces of Lover K is also a set of (same but not
all) t-dimensional vector subepaces of Lover F.
And S is, simultaneously,
a spread of PG(l, qt) and a spread of PG(2t-l, q). - As we shall see later, not
all spreads of PG(2t-l, q) can be obtained in this manner if t > 1.
Before turning to our construction we should like to raise a point which may
Again let ~ be a (finite or infinite) projective space
be of some interest.
of odd dimension 2t-l, but now assume that t
tion,
S, of
(t-l)~dimensional projective
provided that each
is at least two.
subspaces of L a dual spread of
(2t-2)-dimens1onal projective subspace of
and only member of S.
If
~
Call a collec-
~
~
contains one
is finite, it is easy to see that the class of all
spreads of E is identical With the class of all dual spreads of E.
obvious whether the two classes need coincide when E is infinite.
It is not
•
5
4. The construction.
will be valid for
t=l
Let
t
be a positive integer.
but will only be new for
t
~
Let E be a projective space of even dimension
projective subspace of E
spread of E'.
of dimension
2t-1.
What we have to say
2.
2t, and let E'
Furthermore, let 8
be a fixed
be a fixed
We construct a system
(which will turn out to be an affine plane) as follows:
n are the points of E which are not in E I
The points of
The
~
of
intersect E'
11:
•
are the t-dimensional projective subspaces of E which
in a unique member of 8, and are not contained in E'.
n is that induced by the incidence relation of
The incidence relation of
E.
Theorem 4.1.
Corollary.
of order
The system
11:
= 1I:(E, E',
If .E = PG(2t, q)
then
s)
11:
is an affine plane.
=
n(E, E',
S)
is an affine plane
t
q •
Remark.
If
guesian) and E'
n(.E, E I, 8)
t=l, so that
is a projective :plane (not necessarily Desar-
E
is a line of E, then
8
is the set of all points of E'
and
is isormorphic to the affine plane obtained from E by deleting
the line E I
and the :point-set 8.
On the other hand, if
Deaarguesian space and the construction of
Proof.
We may assume
First let P
t > 1,
be a point of
60
11:
t > 1, then .E
is a
n seems to be new.
that E
is a Desarguesian space.
and let J
be a member of 8.
Then, since
J is a {t-l)-dimensional :projective subs:PQCe of E which is contained in E'
and since P
is a point of E which is not contained in .E I , there exists one
and only one t-dimensional projective subspace,
J
and P.
Moreover,
L
n E' =
J.
Hence
L, of lL which contains both
L is a line of
n.
Thus:
there is
•
6
L, of
one and only one line,
member,
J, of
liext let
member,
in common.
L , L
1
S, then, as we have just shown,
That is:
two distin.ct lines of
Next suppose that
are distinct members of S.
2
hence
n
~.
be two distinct lines of
2
J, of S are :Parallel.
J 11 J
containing a given point P
of
n and a given
S.
of
J,
~
If
L , L
1
L , L
1
2
2
contain the same
can have no point of
n which contain the same membe:F.L
(I E 1
L
i
In this case,
= J i (i
= 1,
2), where
(l J
is empty, and
1
2
However, since L , L
are
1
2
J
has no point in common with E t •
2
t-dimensiona1 projective subspaces of the (2t)-dimensiona1 projective space
L
then
1
L
L1n L
contains at least one point,
2
is a point of
S
have at
n.
1ee~t
parallel axiom:
not on
P
Thus:
one common point of
~.
If
~
is a line of
L
th~.q,_t~e
L,
two lines of
Finally, let
P, of E.
~
And P, not being in E I
and if P
is a point of
r~!
is a spread of E "
there exists aline,
R and hence J.
tains
fore
P
L
then P
PQ be.s a unique point,
L, of
and J, then
and
which is
n which contains
~.
Then the line,
PQ, of E
of E.
R, in common with E'.
~
which contains
L
and
contains
Q.
Thus:
L
P
Since
L
S
If
Q, then L must contain
and
is the unique line of
R and hence
is
Since
R is contained in one and only one member, J, of S.
On the other hand, if
contains P
~
in common with L.
P, Q be distinct points of
2t,
j
These two facts lead at once to the
not contained in the (2t..1)-dimensiona1 projective subspace E'
E has dimension
E,
which contain different members of
~
exists one and only one line,
and has no point of
n
contains
PR
n which con...
= PQ.
There-
if P, Q are two distinct points of
Q are contained in one and only one line,
~l
L, ofn.
In order to complete the proof of Theorem 4.1, we need only show that each
line of
obvious.
~
contains at least two distinct points of
~.
But this is sufficiently
•
7
In the case of the Corollary, we need only compute the number,
l-dimensional vector spaces of a (t+l)-dimensional vector space,
which are not in a specified t-dimensional vector subspace,
n( q-l)
whence
Thus, if E
t
cisely n = q
=
J, of
L.
Clearly
t
each line of
n = n(E, E', S)
Since each member"
= neE,
n
p~e-
E', q) has
in a proj ective plane
J, of S
And we adjoin the spread,
S,to
Hence the corresponding projective plane
n*
corresponds to a class of
n, namely those containing J" we adjoin each such J
as a "point at infinity".
infinity".
L, over GF(q),
This proves the Corollary.
We may imbed the affine plane
parallel lines of
of
- q
= PG(2t.. q) .
distinct points.
in the familiar manner.
t+l
q
n,
to
n
n as a "line at
n* has a perfectly concrete
representation in terms of our construction.
If (as will turn out to be the case) some of our planes
guesian (for
n are not Desar-
t > 1), the main advantage of the present construction is that it
exhibits non-Desarguesian projective planes in the realm of classical Desarguesian
projective geometry.
In particular, the various planes
group of all collineations of E.
now must be answered:
n may be related to the
There is, however, a practical question which
How extensive is the present construction?
In order to
give a complete answer we must relate our 'Work to the known the¢)ry of projective
planes, and for this purpose we must make rather more use of coordinates than we
would choose under other circumstances.
5-
An affine rapresentation of spreads.
and let E'
of Section 2,
be a projective space of odd dimension
Ef = E(W/F) where
vector space over
of E'
Let
F
2t-l.
be a positive integer
Then, in the sense
is a skew-field and W is a (2t)-d1mensional
F as a ring of left operators.
which contain a specified
t > 2
We shall study the spreads
(t-1Ldimensional projective subspace of Ef.
•
8
This Subs:p6ce we shall designate by J(co).
Since J(co) is a t-dimensional vector subs:pace of the (2t)-dimensional vector s1Jace W over F, we may choose an arbitrary basis
of J(co)
over F and complete this to a basis of W over F by adjoining
t
additional basis elements
Thus
,
•
Once the bases have been chosen, we define a one-to-one mapping x -> x'
J(co) upon a (t-dimensional vector) subspace of W by the folloWing rule:
of
If
t
x = I: xi e.
1=1
J.
(5.3)
where the
xi are in F, then
t
Xl
= I:
1=1
Next suppose that J
Then J
xi
ei
is any (t-l)-dimensional projective subspace of I:t.
is a t-dtmensional vector subspace of W.
condition that J(co)
and J
A necessary and sufficient
have no point in COlIlDlon is that J(oo)
of the zero-vector alone:
,
nJ
consist
9
or, equivalently, that
J(OO)
+ J
=
W
(5.5), (5.6) will hold precisely when
We note that the equivalent conditions
each w in W has a unique representation w
and y
is in J.
where the xi
= x + y where x is in
Equivalently, we must have, for each i
are in J(oo)
and the
yi
= 1,
J(oo)
2, ••• , t,
constitute a basis of J.
Moreover,
if xi is in J(oo), then
where the X
are in F. As a consequence, there corresponds to each J skew
ij
to J(oo) a unique matrix X = (X ) of t rows and columns with elements in F
ij
such t.hat J = J(X) where
and
(1
Next let X, Y be two distinct
t
by t
= 1,2, ... ,t).
matrices over F, and set
z = X - Y. The interseCjtion of
J(X)
With J(cn)
~
+ J(Y)
spanned by the t
zi
=
t
~
i=l
vectors
zi' e .
J
J
(i = 1, 2, ... , t).
10
Hence
J(X) + J(Y)
x-
W ¢:::;>
=
Y is nonsingular.
Next let us observe the spaces J ( 0) I 1(1)
o
and the identity matrixl I:
Clearly each two of J(oo)1 J(O)I J(I)
following converse:
If J(OO)I
projective subspaces of
Ell
are disjoint.
of W over F
such that
We wish to establish the
L, M are three mutually disjoint (t-l)-dimensional
and if
ell
g •• ,
over FI there exists one and only set of t
LI
corresponding to the zero matrix l
et
is a preassigned basis of J(oo)
additional basis vectors
L = J(O), M = J(l).
We see this as follows:
Since
M are disjoint, then
L+M = V.
Hence I for each i
=1
for unique elements
are disjoint l the
1
2, ••• 1 t,
ei, b
t
in Land M respectively.
i
elements
must constitute a basis of Mover F.
elements
ments
Since
L(OO)
Since
L(oo)
and M are disjoint, the t
ei must constitute a basis of Lover Fj in addition, the
eil ei must constitute a basis of W over F.
and L
2t ele-
This is the promised
proof.
At this point it should be clear that, in considering spreads of
E',
there
11
is no loss of generality, in terms of the above notation, in limiting attention
to spreads containing J(oo), J(O)
and J(I).
8uch a spread,
8,
must have the
following prope !'ties:
(i)
(ii)
8
contains
J(oo).
Each member of 8, other than J(oo), has the form J(X), where
X belongs to a collection,
C , of matrices of
t
rows and columns with elements
in F, subject to the following conditions:
{i1a.)
c:;
(iib)
If X, Yare d±$tinct matrices in
contains the zero matrix.. 0, and the identity matrix, I.
e..
then the matrix X-Y
is
non-singular.
(iiC)
To each ordered pair of elements
corresponds a (unique) matrix X in
e
of J(oo) with a ~
a, b
such that
0
there
aX = b.
In view of the preceding discussion" we need only explain the condition
X
(iiC) and, in particular the notation a.
(5.11)
where the
i
As
a
t
=
t
E
E
i=l
j=l
a consequence, for each t
a --> aX
by
t
a
i
xij e j
•
matrix X with elements in F, the mapping
is a linear transformation of J(oo)
over F.
Now we may explain (iic) as a maximality condition.
are merely normalization conditions.
members of 8
and if
are in F, then
X
(5.12)
= (Xij )
t
= E ai ei
i=l
a
a
First, if X
have a CO!Il'.\llon point.
Conditions (i) and (:Lia)
Condition (iib) merely ensures that no two
As we shall see, (iiC) has precisely the
effect of ensuring that each point of E is contained in at least one member
•
(and hence in exactly one member) of 8.
Consider a point of E1 .. that is.. a
l-dimensional subspace
F, where, of course,
{w)
of W over
w is a nonzero
12
element of W.
In terms of the mapping defined by (5.~), (5.4),
for a unique pair of elements
If a
~
0
then we want
of J{~).
a" b
w to be in J(X)
HO"Tever, in view of (5.7)"
If a
=0
+ a'
w is in 1I(~).
then
for some (unique)
=b
w
t.
X in
w will be in J(X) precisely when (assuming (5.11»
or (in view of (5.8), (5.12»
precisely when aX = b.
In the section which follows we shall use the present discussion to exhibit
the connection between spreads and the so-called Veblen-Wedderburn systems.
For
this reason we shall not bother to give examples at this point.
6. Affine coordinates for
I:
Let
1C.
t
be a positive integer,
be a (2t)-dimensional projective space, let I:'
projective subspace of I:, and let S
t ~ 2" let
be a (2t-l)-dim.ensional
be a spread of I: '.
We wish to introduce
affine coordinates for the affine plane 1C = n(I:, I:', S) defined in Section
As in Section 2 "Te represent
I:
in the form
I:(VIF)
where
F
4.
is a skew-
field and V is a (2t+l)-dimensional vector space over F as a ring of left
operators.
V over
F.
Then I:' =~(W/F) where
vi is a (2t)-dimensional vector subspace of
Without loss of generality we may give a special role to some (arbi-
trarily chosen) ordered triple, J(~), J(O), J(I)" of distinct members of the
spread S.
of
J(~)
by t
2t
Then we may assume that, in the notation of Section 5,
and other members
J(X),
matrices with properties
elements
e , e
i
1 (i
X
E
S
consists
c; " where C is a collection of
(iia), (iib), (iic).
Here W has
t
a basis of
= 1,2, ••• ,t), and we need only add a single el~t,
e*,
of V which is not in W in order to get a basis of V.
We observe that, in terms of the notation of Section 5, each point of
1C,
that is" each l-dimensional vector space of V over F which 1s not in W, has
a unique basis element of the form
13
y
where
x·
+
e*
+
x, yare in J(co). Thus we may speak of the point (x, y) of " where
we define
(x,
( 6.1)
=
y)
(y +
for every ordered pair x, y
A
+ e*}
Xl
of elements of
J(co).
line of ", that is, a (t+l)-dimensional vector space of V
over F which
intersects W in a member J of S, has the form
J +
(x,
y)
=J
+ (y +
provided (x, y) is one of its points.
(I) l-ines
x
x'
+ e*}
These lines may be divided into tyro types:
= a. If a is in J(co), the point (x,y) of "lies on
the line
J(CO)
if and only if x
(II)
the point
(a, 0) = J(co)
+
+
(a' + e*}
= a.
Lines
y
= xM +
If b
b.
is in
J(co) and if J(M) is in S,
(x, y) lies on the line
J(M)
if and only if y-b + x'
+
(0, b)
is in
=
J(M) + (b + e*}
J(M); that is, if and only if
M
y - b = x
Now we have specified all the points and all the lines of "
and equations, respectively.
For purposes of comparison we wish to go slightly
further and introduce a coordinate ring (R, +, .).
R
= J(co)
To begin With, we take
and we define addition, +, in R to be the addition in
subspace of V).
by coordinates
J(co) (as a
To specify multiplication in R we must specialize a non-zero
14
element of R
= J(oo)
or, equivalently, we must pick a unit point of
3f.
We pick
the unit point
I = (1,1)
(6.2)
where
( .)
1
=
(1
+
l'
+ e*}
is any fixed nonzero element of
in R
= J(oo)
matrix T(x)
in
as follows:
To each x
C- or a unique member
(6.3)
R
= J(oo).
Then we define multiplication
in R there corresponds a unique
J(T(x»
of S
-
such that
x.
And we define
(6.4)
for all
y, x
in R.
It is now easy to verify that
(R" +, .) has the following
(R, +)
is an abelian group with zero,
o•
properties:
(i)
(ii)
(iii)
.)
(R"
is a groupoid.
If R*
is the set of nonzero elements of R, then
loop with identity element
(iv)
(v)
(x+y)z
= xz
If a, b, c
and only element
x
of
(R*, .) is a
1.
+ yz
for
all
x, y, z
in R.
are elements of H, with a
R such that
xa
= xb
f
b, there eXists one
+ c.
The axioms (i) through (v) characterize a so-called Veblen-Wedderburn system
(or quasi-field).
See" for example"
M. Hall [5J" Bruck [6J
However, in the present case we have additional properties.
f(x + y) = fx + f'y
and
(6.6)
(fx)y
=
:r(xy)
orP1ckert [7J.
Obviously
15
for every f
1
in the skew-field F and for all x" y in R.
is the identity element of R" we see from
(fl)x
for all f
in F"
x
in F.
= fl
(6.6) that
fx
in R, and from
(f + g)l
for all f" g
=
In addition" if
(6.5)" (6.6) that
+ gl ,
(fg)l
=
(fl)(gl)
Consequently, the mapping
f -> fl
is an operator-isomorphism of F upon a skew-field Fl which is a subsystem of
(R, +" .). Therefore we may imbed F as a sub-skew-field of
properties
(6.5)" (6.6)
by making the identification
(R, +, .)
= fl
f
with
for every f in F.
At this point we need a knovm lemma:
Lemma 6.1.
Let
(H, +, .) be a Veblen-Wedderburn system and let F be the
(6.5)" (6.6) for all x, y in R.
set of all elements f
in R which satisfy
Then the subsystem
+,.) of (H, +,.) is a skew-field.
Definition.
(F"
We shall call the skew-field F of Lemma
6.1 the left-
operator skew-field of the Veblen-Wedderburn system (H, +" .).
Proof. With each element x of R we associate a mapping, R(x)" of H,
the right-multiplication by x, defined by
for all y in R.
abelian group
a
~ 0"
In view of axiom (iv), each R(x)
(R, +).
is an endomorphism of the
By axiom (iii), if a, b are elements of R with
there exists a unique x in R such that aR(x)
= b.
Hence the set
of right multiplications of R is an irreducible set of endomorphisms of
Therefore" by Schur's
Lemma" the centralizer"
(f!,. *"
of
~in the
~
(R"
+).
ring of all
16
endomorphisms of
a*
(R, +), is a skevT-field.
An endomorphism,
e,
of (R, +) is in
if and only if
(6.8)
(ye)x =
for all y~x in R.
e = fx
x
f
= 1 in (6.8), we get
Setting y
(6.9)
for all x, where
(yx)e
= leo
From
(6.9) in (6.8), we get
(fy)x = f(yx)
(R, +),
morphism of
and if
e ts
satisfies
That is, (6.6) holds.
in R.
for all y, x
In addition, since 9 is an endo-
(6.5) holds. Conversely, if f satisfies (6.5), (6.6)
then
(6.9), then e is an endomorphism of (R,
defined by
(6.8). If, further, g satisfies (6.5), (6.6) and if
~
+) which
is defined
by
xcp
= gx
= xe
+ ~
,
then
x(e + (0)
= fx
+ gx
= (f
+ g)x
and
x(e<P)
=
(Xl;))c~
Consequently the system (F, +, .)
skew-field
=
g(fx)
is
a.
= (gf)x
skevT-field anti-isomorphic to the
@(*.
\
Now we may add our crucial axiom:
(Vi)
The Veblen-Wedderburn system
CR,
+, .)
is a finite-dimensional
vector space over its left-qperator skew-field.
We note that, although the skew-field F from which we started is not
necessarily the full left-operator skew-field of (R, +, .), nevertheless, (R, +)
17
must, a fortiori, have finite dimension over its left-operator skew-field.
Axiom (vi) raises a question to which the answer is probably unknown:
Has
every Veblen-Wedderburn system finite dimension over its left-operator skewfield?
~~
Translation planes.
planes;, see M. Hall [5].
For an axiomatic characterization of translation
We need merely say here that a projective plane
is a translation plane with respect to one of i.ts lines,
corresponding affine plane
can be coordinatized by a Veblen-Wedderburn system.
to speak of the affine plane
L , if and only if the
obtained from n* by deleting
~,
L and its points,
It will be convenient here
as an affine translation plane.
~
n*
Now we may
state a theorem:
Theorem 7.1.
Every affine plane
is a translation plane.
~(E, EI,
Conversely, if
~
S), constructed as in Section 4,
is alLe,t'fine trenslation plane with
a coordinating Veblen-Wedderburn system which is finite dimensional over its leftoperator skew-field, then ~
is isomorphic to at least one plane
~(E,
E', S).
I
Corollary. Every finite affine translation plane is isomorphic to at least
one plane
~(E,
Proof.
7.1.
We need only concern ourselves with the second sentence of Theorem
Suppose then that
(R, +, .).
that
E', S).
~
is coordinatized by a Veblen-Wedderburn system
Suppose also that
(R, +, .)
has a sUbsystem,
F
= (r,
+, .), such
F is a skeW-field contained in (but not necessarily equal to) the left-
operator skew-field of
space over F, where
points of
~
(R, +, .) and such that R is a t-dimensional vector
t
is a positive integer.
are ordered pairs
long to two types: (I)
the lines
It is to be understood that the
(x, y), x, Y € R, and that the lines of
x
= a,
one for each a
~
is indeed an affine plane.
be-
in R; (II) the lines
y = xm + b, one for each ordered pair of elements m, b in R.
the verification that
~
We shall omit
18
It will be convenient to
F.
s~eak
of R as a t-dimensional vector
Next we introduce a second vector space
isom.o~hic
R',
but having only the zero vector in common with
R.
s~ace
to Rover
over
F,
Then we define the vector
space
W = R +
,
R'
the direct sUIDof Rand R', whose zero element coincides with that of
R'.
We shall understand tha.t the
Rand
ma~p1ng
x -> x'
is an isom.o~hism of
tor space over F.
R upon
R'
over F.
Then W is a (2t)-dimensional vec-
We define
=
J(crJ)
and, for each element
m of
R
R, we define J(m)
to be the set of all vectors
of form
xm + x'
where
F
ranges aver R.
x
Then J(m)
is a t-dimensional sUbspace of
vi over
and
for each m.
Indeed,
xm + x'
is in J(crJ)
J(m)
n
J(k)
=R
=
if m ~ k
[O}
for if
:lOll
+ x'
=
yk + y'
then
xm - yk
whence
y
=x
= (y
and
O=xm-xk
- x)'
=
x
if and only if
0
,
= O.
Similarly,
19
Since m I k l the latter equation has the unique solution x
w
=
y
+ x'
be an arbitrary nonzero element of W.
I
= O.
Next, let
= 01
W
0
If ( and only if)
x
is in
J(co) = R.
On the other handl if x ~ 0, there is one and only one m in
such that
xm = y;
and, for this (and only this) choice of m,
Consequently, if S
m
is the collection consisting of J(co)
w is in J(m).
and the J(m),
R, then S is (in vector form) a spread.
€
Next we introduce a l-dimensional vector space
o
R
[e*}
over
F, baving only
in common with W, and define
v =
W + {e*}
where the direct sum is understood.
1
Thus V is a (2t+l)-dimensional vector
space over F.
At this point, we define
projective space, take
E = E(V IF)
= E(w/F)
E'
to be the usual (2t)-dimensional
to be the corresponding (2t-l)-dimensional
projective subspace of E, and use the spread S, just defined, as a spread of
E I.
It is now a simple matter to verify that the affine plane
isomorphic to the plane
re
from which we started.
This
re(E, E t, S) is
completes the proof of
Theorem 7.1Clearly we have made very little use, in the foregoing proof, of the fact
that
R has finite dimension
the projective s:pace
the problem:
projective
E
t
over
F.
However, without this restriction,
is infinite dimensional, and so is
E'.
And now we see
How do we define the concept of a spread of an infinite-dimensional
~pace?
For specific examples of Veblen-Wedderburn systems - and hence for examples
of spreads - see
M. Hall
[5].
20
8. Some geometric examples.
In order to avoid giving the impression that
the only way to construct spreads is by using
Veble~~vJedderburn systems,
we
shall sketch briefly, without proof, part of a geometric theory developed by
Bruck for spreads of PG(3, g).
We recall that a spread of PG(3, g) is a set of
1 + g2
skew lines of
PG(3, g).
If A, B, C are three distinct skew lines, the set,
taining
A, B, C and
tained in predsely
spreads
D.
As a
(q2 -g}!2
4)p
c>o~seguel1ce, thI:~:...~3.~~lnes
g=2
three skew lines
)
in
is exceptional.
A,B,C
t,
of transversals
A!f3;0
~~e
con=-..
c1j 8tinct regular spTea,lls, anri eB.ch t'Wo of these
h~ve rr8cisel~~~~4,B?C
The case
rlX-
of PG(3,2)
the rest of the discussion we assume
c~~
Every spread of PG(3,2) is regular; and
are contained in precisely one spread.
g
> 2.
In
21
S be a spread of PG(3" q) "
Let
c;J2 •
If S r
regulus
rJ?'"
q > 2" which happens to contain a regulus
is derived from S by replacing the regulus cdPby the opposite
then Sf
the other is not.
is a spread.
Moreover" if one of S,
S'
is regular,
It seems reasonable to conjecture that every spread may be
obtained from a regular spread by iteration of the process of replacing a regulus
by the opposite regulus.
Next suppose that
a quadric
~.
c;i2, cR.,
The conjecture is correct for
(j!,
a
f
are opposite reguli of PG(3" q), belonging to
It may be shown that if S, Sf
respectively, then
q = 3.
sns,
are reaular spreads containing
consists of precisely two lines
A, B,
namely of a pair of conjugate non-secants of the quadric:;L •
Again, let A, B be two arbitrarily chosen lines of a regular spread S
of PG(3,q).
titioned into
It may be shown that the remaining
q-l
disjoint reguli
by the requirement that, for each
{2 i with reguli c,fi'
one of (;i!i' cJei for each
quadric
with
which are non-regular for
2
are
&1'
i,
~,
(l-l
lines of S are J2ar-
..., ~-l
uniquely defined
A, B are conjugate non-secants to the
ai.
As a consequence, by combining A, B
q l
i, we get 2 distinct spreads, many of
q large.
If
q = 3, there are
4 spreads, of which
regular and 2 are non-regular.
The analYtic formulation of the above remarks is quite interesting.
the notation of Section 6, with F = GF( q)
"\the
and t=2, except that, to emphasize
fact that we are dealing with lines, we use
L instead of J.
(8.1)
,
(8.2)
vThere
We use
X is a matrix of two rows and columns over GF( q).
Then
22
If
of
a
(j( is the regulus determined by
L(oo)
E
L(T)
and the lines
GF(q).
L(aI)
L(oo), L{O),
T
Equivalently,
ranges over
T
q2
ranges over
then
~consists
corresponding to the scalar matrices
Each regular spread containing
where
tfl),
a
consists of
L(oo)
and lines
matrices forming a field isomorphic to GF(q2).
q2 matrices of form
aI + bX
,
a, b E GF(q)
where X is a fixed (but arbitrarily chosen) irreducible matrix.
of spread consists of
r.2f! and
aI"
q2_ q lines
L(Y)
where
Another type
Y ranges over the
q2. q distinct conjugates
of a fixed (but arbitrarily chosen) irreducible matrix X.
not regular, but becomes regular when (£is replaced by
The latter spread is
(2!f2
t.
Of the two types of spread just described, the regular spread corresponds to
a Veblen-Wedderburn system which is a field, and hence corresponds (according to
our construction) to a Desarguesian plane.
On the other hand, the non-regular
spread corresponds to a Veblen-vJedderburn system which is a Hall system, and
hence corresponds (according to our construction) to a Hall plane.
Returning again to the regular spread defined above, we may verify that
the
q-l
reguli
a?,.."., ~
~,
q-l
of the spread, with respect to which
L{oo), L(O)
are conjugate non-secants, consist, for each
L{aI + bX)
such that the determinant
~
laI + bX I
Clearly, if q is even,
e~ch ~
has a constant value
d
line in common with
(J2,
line in common with
rR and the other half are disjoint from cJ2 .
i
O.
1, of the lines
while, if
has a unique
q is odd, half of the@(o
J.
have a unique
So far we have no satisfactory formulation of a geometric theory of spreads of
(2t-l)-dimensional projective space for the case
t > 2.
23
9. A connection with a procedure of Ostrum. In a series of papers, of
which we shall mention only two, Ostrum [8, 9J has developed a procedure for
constructing from a given plane
of finite order
1C
which is symmetrically related to
11"
n
2
a "conjugate" plane
but usually has different properties..
1C
We explain the procedure in its affine form:
Let
K, made up of all the lines of some
that, to every pair,
a line,
P, Q, of distinct points of
(PQ)'
are all in K.
of the sUbplanes
(PQ) • •
the points of
and the lines of
PQ
1C'
1C
Let
1C'
1C
of
such that the line
1C',
(PQ)f
1C'
wi th
be the collection consisting
1C
by retaining
2
n.
1C';
Moreover,
K'
consists of all the
and K is a collection of affine
P, Q of distinct points of
is in K'.
Now let us apply Ostrum's procedure to an affine translation plane
1C
=
1C ( I:, I:
r,
S)
the projective
of order
2
n=
q2
constructed by the method of Section 4 from
4-space I: = pG(4, q).
Here I:' = PG(3, q)
space of I: and S, a spread of I:', consists of
recall that the points of
that the lines of
in a line
of S.
1C
1C
n;
in K' - and using the latter as lines.
one for each pair,
of
by
which are not in K but replacing the
n+l parallel classes of
n
1C
Q; (ii) (PQ) , has order
be the system obtained from
is an affine plane of order
subplanes of order
and
Let K'
in K by the subplanes (PQ)'
lines of some
1C'
which are joined in
1C
such
1C,
(There can be at most one Bubplane
(PQ)' .vttaprqlerties (i), (ii), (iii).)
Then
1C
PQ, in K, there corresponds an affine subplane, (PQ)', of
(iii) the lines of
lines
possessing a collection,
n+l parallel classes of lines of
the following properties: (i) (P-Q)' contains P
e
n2
be an affine plane of finite order
1C
q2+1
is a projective 3-
skew lines of I:'.
are the points of I: which are not in I:'
are the planes of I: which are not in I:'
We
and
but meet I: ~
24
First, consider any plane,
in a line of S.
~,
L corresponds a set,
Let
~,
(T)
order
q.
Then S meets
C, of
~
~t
q(q+l)
(T).
Hence
planes of
~
in
a line,
~t
and does not meet
L, which is not in S.
To
q2 points of T which are not in
containing a line of C and at least one
It is easy to see that
1f
which is not in
q+l. lines of S, one through each point of L.
be the system consisting of the
and of the
point of
T, of
(T)
is an affine sUbplane of
has affine subplanes of order
1f
of
q in rich profusion.
Next suppose that the spread S happens to contain a
regulus,~.
In
--.,
the sense of Ostrum's procedure, let K be the collection consisting of all
planes of ~ which are not in ~,
and which meet ~ I
K consists of all the lines of some
in a line of ()::( • Then
q+l parallel classes of lines of
1f.
And - in view of the preceding paragraph - Ostrum's procedure amounts, in this
case, to changing the spread S by replacing the regulus
regulus
~ t.
c;J! by the opposite
In particular" the conjecture about spreads of PG(:;,q) men-
tioned in Sectton 8 could be rephrased as a conjecture that, by iteration of
Ostrum's procedure, any translati<m plane obtainable by our construction from
r
PG(4, q)
f
could be derived from a Desarguesian plane.
BIBLIOGRAPHY
1.
R. H. Bruck.
Existence problems for classes of finite planes.
Mimeographed notes of lectures delivered to the Seminar of the Canadian Mathematical Congress, meeting in Saskatoon, Sask., August 196;.
2.
O. Veblen and J. H. M. Wedderburn.
geometries.
Trans. Amer. Soc.
;a. C. Radhakrishna Rao.
theory of numbers.
52 (1907), ;79-;88.
Finite geometries and certain derived results in
Proc. Nat. Inst. Sci. India,
;b.
11
(1945), 1;6-149.
• Difference sets and combinatorial arrangements
derivable from finite geometries.
4.
R. Baer.
5.
Marshall Hall, Jr.
6.
R. H. Bruck.
geometry.
Non-Desarguesian and non-Pascalian
Proc. Nat. Inst. Sci. India, 12 (1946), 12;-1;5.
Linear Algebra and Projective Geometry.
The Theory of Groups.
New York, 1952.
New York, 1959.
Recent advances in the foundations of Euclidean plane
AIDer. Math. Monthly 62, No.7, II. (Slaught Memorial Paper No.4),
(1955), 2-17.
7.
G. Pickert.
Projektive Ebene.
8. T. G. Ostrum.
Berlin-Gottingen-Heidelberg, 1955.
Semi-translation planes. Trans. Amer. Math. Soc.
To
Nets with critical deficiency.
To
IiLppear.
9.
appear.
T. G. Ostrum.
Pacific J. Math.