The geometric structure and the p-rank of an affine triple system derived from a nonassociative moufang loop with the maximum associative center

AbstractH. P. Young showed that there is a one-to-one correspondence between affine triple systems (or Hall triple systems) and exp. 3-Moufang loops (ML). Recently, L. Beneteau showed that (i) for any non-associative exp. 3-ML (E, · ) with ‖E‖ = 3n, 3 ⩽ ‖Z(E)‖ ⩽ 3n−3, where n ⩾ 4 and Z(E) is an associative center of (E, ·), and (ii) there exists exactly one exp. 3-ML, denoted by (En, ·), such that ‖En‖ = 3n and ‖Z(En)‖ = 3n−3 for any integer n ⩾ 4. The purpose of this paper is to investigate the geometric structure of the affine triple system derived from the exp. 3-ML(En, ·) in detail and to compare with the structure of an affine geometry AG(n, 3). We shall obtain (a) a necessary and sufficient condition for three lines L1, L2 and L3 in (En, ·) that the transitivity of the parallelism holds for given three lines L1, L2 and L3 in (En, ·) such that L1‖L2 and L2‖L3 and (b) a necessary and sufficient condition for m + 1 points in En (1 ⩽ m < n) so that the subsystem generated by those m + 1 points consists of 3m points. Using the structure of hyperplanes in (En, ·), the p-rank of the incidence matrix of the affine triple system derived from the exp. 3-ML(En, ·) is given