从非平移射影平面看素数幂猜想

射影平面是指符合任意两个点在唯一一条线上，和任意两条线交于唯一一个点的设计组合。若射影平面只有有限个点，则其每条线也会有同一数量的点，该平面的阶是这数量减一。已知的有限射影平面的阶皆是素数幂。故有猜想：有限射影平面阶数是素数幂。若射影平面上存在一条线L，使得能逐点固定L的自同构子群是传递地作用于不在L的点集上，则称这个平面为平移平面，其阶必为素数幂。学界对于阶可能是非素数幂的非平移平面知之较少，这阻碍了研究素数幂猜想。故认识和构作非平移平面是基本核心问题。..本项目将深入研究非平移平面，并把其推广以构作新的平面，及通过寻找阶不是素数幂的平面以增进对素数幂猜想的了解。我们会以锥体表示法和域减法等方法研究已知平面，包括休斯平面、菲氏平面和零星平面，了解其子线、子平面在不同表示法下的结构和分布，以高维几何和代数重新描述平面及用以构作新平面。我们预期能对构作非平移平面和研究素数幂猜想有显著进展。

A projective plane is a point-line incidence structure with the following properties: any two points lie on a unique line; any two lines intersect at a unique point. In a finite projective plane, every line has the same number of points. The order of the plane is defined to be this number minus one. Every known finite projective plane has a prime power order. It is a famous long-standing problem: the conjecture that the order of any finite projective plane is a prime power. A projective plane is called a translation plane if there exists a line L such that the automorphism group fixing L pointwise acts transitively on the points not on L. Every translation plane has a prime power order. There is less known about non-translation planes, which hinders solving the prime power conjecture. Thus it is a fundamental and core problem to study and construct new non-translation planes...This project will focus on studying known non-translation planes, on constructing new non-translation planes, and on advancing knowledge about the prime power conjecture by looking for planes of non-prime power order. To achieve this, we shall investigate the structure and distribution of the sublines and subplanes of the known non-translation planes, such as the Hughes planes, the Figueroa plane and sporadic planes, under the cone-representation and field reduction. This will give alternative descriptions of these planes in terms of higher dimensional projective spaces or related algebra. We then generalize the descriptions to construct new planes. We expect to have significant progress towards the construction of non-translation planes and solving the prime power conjecture.