Ranks of Nets

Let N be a k-net of prime order p. We find bounds on the p-rank of (the point-line incidence matrix of) N for k ∈ {3, 4}, and observe connections between the p-rank and certain structural properties of N . Implications for the study of finite projective planes are described. 1. Loops and 3-Nets of Prime Order Let (L, ∗) be a loop of prime order p. The 3-net N = N (L) coordinatized by L is the point-line incidence system having p points L = L× L, and 3p lines given by {a} × L for a ∈ L (the lines “x = a”); L× {b} for b ∈ L (the lines “y = b”); and {(x, y) ∈ L : x ∗ y = c} for c ∈ L (the lines “x ∗ y = c”). The point-line incidence matrix of N is the p2×3p matrix with rows and columns indexed by points and lines of N respectively; and having entries 0 and 1 corresponding to nonincident and incident point-line pairs respectively. We have 1.1 Main Theorem [5]. The p-rank of the incidence matrix of N equals 3p−3 if L is associative, and 3p−2 otherwise. Our original proof [5], still the simplest proof available, uses loop theory. (Here for simplicity we consider only loops and nets of prime order, although more arbitrary finite orders were considered in [5].) We reproduce this proof below; and we indicate three alternative proofs of the same result. Our (currently unrealized) goal is a generalization of Theorem 1.1 to k-nets for k = 3, 4, . . . , p+1; possibly using techniques from nonassociative algebra, or possibly by generalizing some of the other techniques described in this paper. The desired generalization of this result is 1.2 Conjecture [5]. Let N be any k-net of prime order p, and let N ′ be any (k−1)-subnet ofN obtained by deleting one of the k parallel classes of lines ofN ; here k ∈ {2, 3, . . . , p+1}. Then the p-rank of the incidence matrix of N exceeds that of N ′ by at least p−k+1. The significance of Conjecture 1.2 lies in the fact [5] that this would imply that every projective plane of prime order is Desarguesian, thereby settling one of the most celebrated currently open problems in finite geometry. Extensions of this method to other finite orders would yield restrictions on the possible orders of finite projective planes, beyond the restrictions available through the Bruck-Ryser Theorem [2]. We believe that these finite geometric questions are worthy of the attention of researchers in nonassociative algebra. Indeed, Belousov [1] attributes the origins of quasigroup theory to the study of