On Projective Planes of Order 12 Which Have a Subplane of Order 3, I
On Projective Planes of Order 12 Which Have a Subplane of Order 3, I
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在具有 3 阶子平面的 12 阶射影平面上，I
DOI：
10.1016/0097-3165(80)90017-5
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发表时间：
1980
期刊：
影响因子：
--
通讯作者：
T. Trung
中科院分区：
文献类型：
--
作者：
Z. Janko;T. Trung
证据。设P是一个12阶射影平面，它有一个3阶子平面PO，r是13阶PO的任意自同构.P的点i，2，…，13(整数mod 13)可记为x‘=x+1(对所有点x)，即r=(1，2，…，13).我们称P的点为“原点”或“天”，称P的直线为“原线”或“天”.每条原始线都包含(与)4个原始点和P中的9个其他点，我们称之为“新点”。我们正好有13 9=117个新点数。因此，恰好还有27个P点不与任何原始直线重合。我们称这27分为“女生”，并用co，0，L，…，25表示。任何包含至少一个女学生的P线都被称为“学校线”。取一个原点x。通过x恰好通过4条原始线和9条其他线。设m是通过x的任何不是原始线的固定线。恰好有9条原始线不通过x，m必须与它们相交于9个不同的新点。因此，在m上只剩下3个点，它们不与任何原始直线相关联。由此推论，m正好包含3个女学生，因此，特别是m是一条校线。从x到x的9条非原创线路是学校线路，每条线路正好包含3名女学生。因此，这9条校线将27名在校女生分成9组，每组3人，这样就构成了x日的“学校游行”。
Proof. Let P be a projective plane of order 12 which has a subplane PO of order 3 and let r be any automorphism of PO of order 13. The points I, 2,..., 13 (integers mod 13) of P, can be so denoted that x’= x+ 1 (for all points x), ie, r=(1, 2,..., 13).We call the points of P,,“original points” or “days” and the lines of P,“original lines” or “days out.” Every original line contains (is incident with) 4 original points and 9 other points in P which we call “new points.” We have exactly 13 9= 117 new points. Hence there remain exactly 27 points of P which are not incident with any original line. We call those 27 points “school girls” and denote them with co, 0, l,..., 25. Any line of P which contains at least one school girl is called a “school line.” Take an original point x. Through x pass exactly 4 original lines and 9 other lines. Let m be any fixed line through x which is not an original line. There are exactly 9 original lines which do not pass through x and m must intersect them in 9 distinct new points. Hence there remain exactly 3 points on m which are not incident with any original line. It follows that m contains exactly 3 school girls and so in particular m is a school line. The 9 nonoriginal lines through x are school lines each containing exactly 3 school girls. Therefore these 9 school lines partition the set of 27 school girls into 9 sets of 3, making in this way a “school parade” of the day x. Conversely, let