A note on some partially transitive projective planes
A note on some partially transitive projective planes
复制标题
关于部分传递射影平面的注释
DOI：
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发表时间：
1957
期刊：
影响因子：
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通讯作者：
D. Hughes
中科院分区：
文献类型：
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作者：
D. Hughes
1.导论.在[3]中，作者提出了m = 3，n ^4的“（4，m）型”射影平面的存在性问题。利用Baer和Paige [1; 4]的结果，我们将证明这样的平面不存在。另一方面，我们将证明对于每个n = p2 t，p为素数，确实存在（4，m）型平面。对于背景材料和定义，读者可以参考[3]和那里给出的参考文献;特别是，平面三元环的概念在下文中将是重要的。然而，我们简要地定义了（4，m）类型的投影平面类。假设它是一个（有限）阶的射影平面re，并且是ir的一个直射群，其中&固定（按元素）线Ko上的mS 3点Qit i= 1，2，···，m，不在KQ上的点Qo，以及线K 0和Ki = Q 0 Qi，*=1，2，· · ·，m。普通点（线）是不在任何线Kj上的点（不包含任何点Qi的线），i = 0，1，···，m;进一步假设在普通点和普通线上是传递的和正则的。则它是（4，m）型的。中显示
1. Introduction. In [3] the author raised the question of the existence of a projective plane of "type (4, m)" with m = 3, n ^4. Using results of Baer and Paige [l; 4], we shall show that such a plane does not exist. On the other hand, we will show that a plane of type (4, m) does exist for every n = p2t, p a prime. For background material and definitions the reader is referred to [3] and the references given there; in particular, the notion of a planar ternary ring will be important in what follows. However, we briefly define the class of projective planes of type (4, m). Suppose it is a projective plane of (finite) order re and ® is a collineation group of ir, where & fixes (element-wise) mS3 points Qit i=l, 2, ■ ■ ■ , m, on a line Ko, a point Qo not on KQ, and the lines K0 and Ki = Q0Qi, *=1, 2, • • • , m. An ordinary point (line) is a point not on any of the lines Kj (a line not containing any of the points Qi), i = 0, 1, • ■ • , m; suppose furthermore that ® is transitive and regular on the ordinary points and ordinary lines. Then it is of type (4, m). It was shown in