Divisible designs, Laguerre geometry, and beyond

Von Staudt’s theorem revisited

“…Hence the bases (e 0 , e 1 ) and (e ′ 0 , e ′ 1 ) satisfy (13) so that Lemma 2 can be applied to them (without any notational changes). We claim that α, as defined via (27), coincides with the Jordan homomorphism β appearing in Lemma 2: Indeed, α satisfies the defining equation (14) according to the second row of table (28) in conjunction with the induction hypothesis. We now introduce bases ( f 0 , f 1 ) of M and ( f ′ 0 , f ′ 1 ) of M ′ as in Lemma 3, but replace the arbitrary t ∈ R from there by the given t 2 ∈ R. This gives a second table of coordinates:…”

Linear sets in the projective line over the endomorphism ring of a finite field

“…Any two distinct points of PG(1, E) are distant precisely when they belong to a common F -chain [2, Lemma 2.1]. From this we obtain the following result [8,Thm. 3.4.7], which is a slightly modified version of [2, Thm.…”

“…We therefore can repeat the second part of the proof of Prop. 2.10 up to (8). By this formula 3 , there exists an automorphism η ∈ Gal(F q t /F q ) and an invertible matrix (c ij ) ∈ GL 2 (q t ) such that…”

“…Take notice that the elements cij that are used now play the role of the elements dcij that appear in(8).…”

Affine and projective planes linked with projective lines over certain rings of lower triangular matrices

“…, n. The projective line P(T n (q)) over the ring T n (q) is the set of all cyclic submodules T n (q) X, Y , where X, Y ⊂ 2 T n (q) is unimodular. We refer to [7,4] for definitions of the unimodularity and the projective line in the case of an arbitrary finite associative ring with unity. It is well known that any unimodular pair generates a free cyclic submodule.…”