In mathematics, André planes are a class of finite

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-Desarg ...

s found by André. The

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 at a single point, but there are some pairs of lines (namely, parallel lines) that d ...

and the

Hall plane In mathematics, a Hall plane is a non-Desarguesian projective plane constructed by Marshall Hall Jr. (1943). There are examples of order ''p''2''n'' for every prime ''p'' and every positive integer ''n'' provided . Algebraic construction via Hal ...

s are examples of André planes; the two-dimensional regular nearfield planes are also André planes.

Construction

Let

F = GF(q)

be a finite

field Field may refer to: Expanses of open ground * Field (agriculture), an area of land used for agricultural purposes * Airfield, an aerodrome that lacks the infrastructure of an airport * Battlefield * Lawn, an area of mowed grass * Meadow, a grass ...

, and let

K = GF(q^n)

be a degree

n

extension field In mathematics, particularly in algebra, a field extension is a pair of fields K \subseteq L, such that the operations of ''K'' are those of ''L'' restricted to ''K''. In this case, ''L'' is an extension field of ''K'' and ''K'' is a subfield of ...

F

. Let

\Gamma

be the group of

field automorphism In mathematics, an automorphism is an isomorphism from a mathematical object to itself. It is, in some sense, a symmetry of the object, and a way of mapping the object to itself while preserving all of its structure. The set of all automorphisms ...

s of

K

over

F

, and let

\beta

be an arbitrary mapping from

F

\Gamma

such that

\beta(1)=1

. Finally, let

N

be the

norm Norm, the Norm or NORM may refer to: In academic disciplines * Normativity, phenomenon of designating things as good or bad * Norm (geology), an estimate of the idealised mineral content of a rock * Norm (philosophy), a standard in normative e ...

function from

K

F

. Define a quasifield

Q

with the same elements and addition as K, but with multiplication defined via

a \circ b = a^ \cdot b

, where

\cdot

denotes the normal field multiplication in

K

. Using this

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 ...

to construct a plane yields an André plane.

Properties

# André planes exist for all proper prime powers

p^n

with

p

prime and

n

a positive

integer An integer is the number zero (0), a positive natural number (1, 2, 3, ...), or the negation of a positive natural number (−1, −2, −3, ...). The negations or additive inverses of the positive natural numbers are referred to as negative in ...

greater than one. # Non-Desarguesian André planes exist for all proper prime powers except for

2^n

where

n

is prime.

Small Examples

For planes of order 25 and below, classification of Andrè planes is a consequence of either theoretical calculations or computer searches which have determined all translation planes of a given order: * The smallest non-Desarguesian André plane has order 9, and it is isomorphic to the

of that order. * The translation planes of order 16 have all been classified, and again the only non-Desarguesian André plane is the

. * There are three non-Desarguesian André planes of order 25. These are the

, the regular nearfield plane, and a third plane not constructible by other techniques. * There is a single non-Desarguesian André plane of order 27. Enumeration of Andrè planes specifically has been performed for other small orders:

References

{{DEFAULTSORT:Andre plane Finite geometry