Bruck nets,codes, and characters of loops |
| |
Authors: | G. Eric Moorhouse |
| |
Affiliation: | (1) Department of Mathematics, University of Wyoming, 82071 Laramie, WY, USA |
| |
Abstract: | Numerous computational examples suggest that if k-1 k are (k- 1)- and k-nets of order n, then rankp k - rankp k-1 n - k + 1 for any prime p dividing n at most once. We conjecture that this inequality always holds. Using characters of loops, we verify the conjecture in case k = 3, proving in fact that if pe n, then rankp 3 3n - 2 - e, where equality holds if and only if the loop G coordinatizing 3 has a normal subloop K such that G/K is an elementary abelian group of order pe. Furthermore if n is squarefree, then rankp = 3n - 3 for every prime p ¦ n, if and only if 3 is cyclic (i.e., 3 is coordinated by a cyclic group of order n).The validity of our conjectured lower bound would imply that any projective plane of squarefree order, or of order n 2 mod 4, is in fact desarguesian of prime order. |
| |
Keywords: | |
本文献已被 SpringerLink 等数据库收录! |
|