Let $\mathfrak{d}$ be the Lie superalgebra of superderivations sheaf sections exterior algebra homogeneous vector bundle $E$ over flag variety $G/P$, where $G$ is a simple finite-dimensional complex group and $P$ its parabolic subgroup. Then, transitive irreducible whenever defined by an $P$-module $V$ such that highest weight $V^*$ dominant. Moreover, simple; it isomorphic to fields on superpo...

If F is a geometry for this rank 3 diagram and F' is its shadow geometry with respect to the central node of the diagram, then f satisfies the Intersection Property (IP), it belongs to the following diagram where L and L* denote the classes of linear spaces and dual linear spaces, respectively. According to a well established habit, we only consider residually connected and firm geometries. Tha...

For any odd prime power q, all (q&q+1)th roots of unity clearly lie in the extension field Fq6 of the Galois field Fq of q elements. It is easily shown that none of these roots of unity have trace &2, and the only such roots of trace &3 must be primitive cube roots of unity which do not belong to Fq . Here the trace is taken from Fq6 to Fq . Computer based searching verified that indeed &2 and ...

Relatively few finite non desarguesian flag-transitive affine planes are known whose collineation groups are solvable. With a single exception (see below), all of the known ones of odd order fall into two families studied in [Ka, Su l ] ; those references also contain some historical remarks. The purpose of this note is to construct an additional family of such planes, explain why they are new,...

At least [6 n&1 1 (q i+1)] qn 2m1 log2 q of these planes have kernel GF(q). Moreover, we construct 7(mi)6i q mi planes, where the sum is over all sequences (mi) behaving as in the theorem; and then we settle the isomorphism problem for these planes (Theorem 5.2). The proof combines methods in [Wi] and [Ka]: these planes arise from symplectic and orthogonal spreads combined with changing from fi...

In this paper, we present a classification of $$\mathbf{2}$$ -designs with $$\mathbf{gcd}({{\varvec{r}}},{\varvec{\lambda }})=\mathbf{1}$$ admitting flag-transitive automorphism groups. If $${{\varvec{G}}}$$ is group non-trivial -design $$\mathcal {{\varvec{D}}}$$ , then either $$(\mathcal {{\varvec{D}}},{{\varvec{G}}})$$ one the known examples described in or has $${{\varvec{q}}} = {{\varvec{p...

In this paper, we consider finite flag-transitive affine planes with a solvable automorphism group. Under a mild number-theoretic condition involving the order and dimension of the plane, the translation complement must contain a linear cyclic subgroup that either is transitive or has two equal-sized orbits on the line at infinity. We develop a new approach to the study of such planes by associ...