Cyclic affine planes of even order

✍ Scribed by K.T. Arasu

• Publisher: Elsevier Science
• Year: 1989
• Tongue: English
• Weight: 446 KB
• Volume: 76
• Category: Article
• ISSN: 0012-365X
• DOI: 10.1016/0012-365x(89)90317-8

✦ Synopsis

In this paper, we prove the following theorem: Suppose there exists a cyclic afhne plane of even order n.

Then (a) either n = 2 or n = 0 (mod 4), and (b) for each prime divisor p of n, we have either (p/q) = 1 for each prime q 1 n2 -1 or for some positive integer r (which depends on p), n+lIp'+landn-lip'-1, according as exp,t_,(p) is odd or even. For p = 2, the former condition cannot hold and hence the latter one holds making exp,+,(2) even. As a corollary, we prove that if there exists a cyclic afline plane of order n = 4 (mod 8), then (i) n must be a square, (ii) n = 1 (mod 3) and (iii) each prime divisor of n + 1 is ~1 (mod 4).

(For an integer a, if t is any integer with (t, a) = 1, exp,(t) would mean the smallest positive integer I such that t' = 1 (mod a).