Prime power conjecture

This article is about a conjecture in the following area in/related to group theory: incidence geometry. View all conjectures and open problems

Statement

The conjecture has the following equivalent formulations:

1. Given a finite Projective plane (?) (viz., a projective plane with only finitely many points), its order (which is defined as one less than the number of points on any line) must be a Prime power (?).
2. The set of nontrivial prime powers (i.e., prime powers other than the number $1$) is precisely the set of numbers that can be realized as the orders of finite projective planes.
3. Given a finite Affine plane (?) (viz., an affine plane with only finitely many points), its order (which is defined as the number of points on any line) must be a prime power.
4. The set of nontrivial prime powers (i.e., prime powers other than the number $1$) is precisely the set of numbers that can be realized as the orders of finite affine planes.

Note that by the definition of projective plane and affine plane, the order of a projective plane or an affine plane cannot be $1$, hence we exclude $1$ from the set of possible orders when interpreting these statements.

Partial truth

It is known that every nontrivial prime power occurs as the order of a finite projective plane, so the question is whether for every number that is not a prime power, there is no projective plane of that order.

For specific numbers

The result is known for the case $n=6$. In other words, there is no projective plane of order six. (Status for $n=10$ and $n=12$?)

General results

The following general results are true:

• Bruck-Ryser theorem
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  -- with suitable transitivity assumptions, only have prime powers