On finite projective planes with a single (P,I) transitivity (Q1102969)

Let \(\pi\) be a projective plane of order n coordinated by a linear ternary field T which is a Cartesian group (i.e., the additive loop of T forms a group). A linear representation \(\rho\) of the additive group of T yields an \(n\times n\) generalized Hadamard matrix with entries in \(\rho\) (T). The existence of two generalized Hadamard matrices of order 12 (one due to Seberry, the other due to the author), though they together do not yield a Cartesian group of order 12, in the author's opinion, gives reason to suspect the truth of the conjecture that all Cartesian groups are elementary Abelian.