新的2^(9)阶旗传递仿射平面

A new flag-transitive affine plane of order 2^(9)

有限旗传递仿射平面与很多组合对象(展形、平面函数、半域和线性化多项式等)有着密切联系,因而在过去50多年来受到研究者的广泛关注. Foulser在1964年已经完整地确定了有限旗传递仿射平面的自同构群.如果一个旗传递仿射平面有一个可解自同构群,则称该平面是可解的,否则称它是不可解的.不可解的情形早在20世纪90年代末已经给出了完整的分类,而可解的情形至今难以给出完整的分类.目前所有已知的可解旗传递仿射平面可以分为两类:C-平面和H-平面,其中H-平面只在奇特征的情形下出现.本文的主要贡献是首次构造出2^(9)阶的非C型旗传递仿射平面并确定了其全自同构群.

Finite flag-transitive affine planes have received much attention during the past fifty years because of their connections with other combinatorial objects such as spreads, planar functions, semifields and linearized polynomials. In 1964, Foulser completely determined the automorphism groups of finite flag-transitive affine planes. If a flag-transitive affine plane has a solvable automorphism group, then the affine plane is called solvable.The non-solvable flag-transitive affine planes have been completely classified in the 1990 s. But the complete classification for the solvable case seems far out of reach. All known solvable flag-transitive affine planes can be classified into two types: C-planes and H-planes, where H-planes only occur in the odd characteristic case. In this paper, we construct the first flag-transitive affine plane of order 2;over its kernel F;, which is not of type C and the largest Singer subgroup of the translation complement has order(2^(3)-1)(2^(9)+ 1)/9.