In mathematics, a Hall plane is a
non-Desarguesian projective plane
In mathematics, a non-Desarguesian plane is a projective plane that does not satisfy Desargues' theorem (named after Girard Desargues), or in other words a plane that is not a Desarguesian plane. The theorem of Desargues is true in all projective ...
constructed by
Marshall Hall Jr. (1943). There are examples of order ''p''
2''n'' for every prime ''p'' and every positive integer ''n'' provided ''p''
2''n'' > 4.
Algebraic construction via Hall systems
The original construction of Hall planes was based on the Hall
quasifield In mathematics, a quasifield is an algebraic structure (Q,+,\cdot) where + and \cdot are binary operations on Q, much like a division ring, but with some weaker conditions. All division rings, and thus all fields, are quasifields.
Definition
A qu ...
(also called a ''Hall system''), H of order
for ''p'' a prime. The creation of the plane from the quasifield follows the standard construction (see
quasifield In mathematics, a quasifield is an algebraic structure (Q,+,\cdot) where + and \cdot are binary operations on Q, much like a division ring, but with some weaker conditions. All division rings, and thus all fields, are quasifields.
Definition
A qu ...
for details).
To build a Hall quasifield, start with a
Galois field
In mathematics, a finite field or Galois field (so-named in honor of Évariste Galois) is a field that contains a finite number of elements. As with any field, a finite field is a set on which the operations of multiplication, addition, subtr ...
,
for ''p'' a prime and a quadratic irreducible polynomial
over ''F''. Extend
, a two-dimensional vector space over ''F'', to a quasifield by defining a multiplication on the vectors by
when
and
otherwise.
Writing the elements of ''H'' in terms of a basis <1, λ>, that is, identifying (''x'',''y'') with ''x'' + λ''y'' as ''x'' and ''y'' vary over ''F'', we can identify the elements of ''F'' as the ordered pairs (''x'', 0), i.e. ''x'' + λ0. The properties of the defined multiplication which turn the right vector space ''H'' into a quasifield are:
# every element α of ''H'' not in ''F'' satisfies the quadratic equation f(α) = 0;
# ''F'' is in the kernel of ''H'' (meaning that (α + β)c = αc + βc, and (αβ)c = α(βc) for all α, β in ''H'' and all c in ''F''); and
# every element of ''F'' commutes (multiplicatively) with all the elements of ''H''.
Derivation
Another construction that produces Hall planes is obtained by applying derivation to
Desarguesian plane
In mathematics, a projective plane is a geometric structure that extends the concept of a plane. In the ordinary Euclidean plane, two lines typically intersect in a single point, but there are some pairs of lines (namely, parallel lines) that d ...
s.
A process, due to T. G. Ostrom, which replaces certain sets of lines in a projective plane by alternate sets in such a way that the new structure is still a projective plane is called ''derivation''. We give the details of this process. Start with a
projective plane
In mathematics, a projective plane is a geometric structure that extends the concept of a plane. In the ordinary Euclidean plane, two lines typically intersect in a single point, but there are some pairs of lines (namely, parallel lines) that d ...
of order
and designate one line
as its
line at infinity
In geometry and topology, the line at infinity is a projective line that is added to the real (affine) plane in order to give closure to, and remove the exceptional cases from, the incidence properties of the resulting projective plane. The ...
. Let ''A'' be the
affine plane
In geometry, an affine plane is a two-dimensional affine space.
Examples
Typical examples of affine planes are
*Euclidean planes, which are affine planes over the real number, reals equipped with a metric (mathematics), metric, the Euclidean dista ...
. A set ''D'' of
points of
is called a ''derivation set'' if for every pair of distinct points ''X'' and ''Y'' of ''A'' which determine a line meeting
in a point of ''D'', there is a
Baer subplane containing ''X'', ''Y'' and ''D'' (we say that such Baer subplanes ''belong'' to ''D''.) Define a new affine plane
as follows: The points of
are the points of ''A''. The lines of
are the lines of
which do not meet
at a point of ''D'' (restricted to ''A'') and the Baer subplanes that belong to ''D'' (restricted to ''A''). The set
is an affine plane of order
and it, or its projective completion, is called a ''derived plane''.
Properties
# Hall planes are
translation plane In mathematics, a translation plane is a projective plane which admits a certain group of symmetries (described below). Along with the Hughes planes and the Figueroa planes, translation planes are among the most well-studied of the known non-Desar ...
s.
# All finite Hall planes of the same order are isomorphic.
# Hall planes are not
self-dual.
# All finite Hall planes contain subplanes of order 2 (
Fano subplanes).
# All finite Hall planes contain subplanes of order different from 2.
# Hall planes are
André planes.
The Hall plane of order 9
The Hall plane of order 9 is the smallest Hall plane, and one of the three smallest examples of a finite
non-Desarguesian projective plane
In mathematics, a non-Desarguesian plane is a projective plane that does not satisfy Desargues' theorem (named after Girard Desargues), or in other words a plane that is not a Desarguesian plane. The theorem of Desargues is true in all projective ...
, along with its
dual and the
Hughes plane
In mathematics, a Hughes plane is one of the non-Desarguesian projective planes found by .
There are examples of order ''p''2''n'' for every odd prime ''p'' and every positive integer ''n''.
Construction
The construction of a Hughes plane is based ...
of order 9.
Construction
While usually constructed in the same way as other Hall planes, the Hall plane of order 9 was actually found earlier by
Oswald Veblen
Oswald Veblen (June 24, 1880 – August 10, 1960) was an American mathematician, geometer and topologist, whose work found application in atomic physics and the theory of relativity. He proved the Jordan curve theorem in 1905; while this was lon ...
and
Joseph Wedderburn
Joseph Henry Maclagan Wedderburn FRSE FRS (2 February 1882 – 9 October 1948) was a Scottish mathematician, who taught at Princeton University for most of his career. A significant algebraist, he proved that a finite division algebra is a fi ...
in 1907. There are four quasifields of order nine which can be used to construct the Hall plane of order nine. Three of these are Hall systems generated by the irreducible polynomials
,
or
. The first of these produces an associative quasifield, that is, a
near-field, and it was in this context that the plane was discovered by Veblen and Wedderburn. This plane is often referred to as the nearfield plane of order nine.
Properties
Automorphism Group
The Hall plane of order 9 is the unique projective plane, finite or infinite, which has
Lenz-Barlotti class IVa.3. Its
automorphism group
In mathematics, the automorphism group of an object ''X'' is the group consisting of automorphisms of ''X'' under composition of morphisms. For example, if ''X'' is a finite-dimensional vector space, then the automorphism group of ''X'' is the g ...
acts on its (necessarily unique)
translation line imprimitively, having 5 pairs of points that the group preserves set-wise; the automorphism group acts as
on these 5 pairs.
Unitals
The Hall plane of order 9 admits four inequivalent embedded
unitals. Two of these unitals arise from Buekenhout's constructions: one is ''parabolic'', meeting the translation line in a single point, while the other is ''hyperbolic'', meeting the translation line in 4 points. The latter of these two unitals was shown by Grüning to also be embeddable in the dual Hall plane. Another of the unitals arises from the construction of Barlotti and Lunardon.
The fourth has an automorphism group of order 8 isomorphic to the
quaternions, and is not part of any known infinite family.
Notes
References
*
*
*
*
*
*{{Citation , last1=Weibel , first1=Charles , authorlink=Charles Weibel, title=Survey of Non-Desarguesian Planes , url=https://www.ams.org/notices/200710/tx071001294p.pdf , year=2007 , journal=
Notices of the American Mathematical Society
''Notices of the American Mathematical Society'' is the membership journal of the American Mathematical Society (AMS), published monthly except for the combined June/July issue. The first volume appeared in 1953. Each issue of the magazine since ...
, volume= 54 , issue=10 , pages=1294–1303
Projective geometry
Finite geometry