Let G be an abelian collineation group of order n2 of a projective plane of order n. We show that n must be a prime power, and that the p-rank of G is at least b+ 1 if n = pb for an odd prime p.

Cyclic curves, i.e. curves fixed by a cyclic collineation group, play a central role in the investigation of cyclic arcs in Desarguesian projective planes. In this paper, the genus of a cyclic curve arising from a cyclic k-arc of Singer type is computed.

In [11] and [12] (textbooks for teachers’ training colleges written by B. Pelle) isometry and similarity are defined not in the classical way but as a product. We continue this way of definition refer to the affine transformation and collineation, and study the fundamental theorems of these mappings analogously to those of isometry and similarity. AMS Classification Number: 00A35 (ZDM: G55, G59)

Let Γ′ and Γ be two Grassmannians. The standard embedding φ : Γ′ × Γ→ P is obtained by combining the Plücker and Segre embeddings. Given a further embedding η : Γ′ × Γ→ P′, we find a sufficient condition for the existence of α ∈ Aut(Γ) and of a collineation ψ : P→ P′ such that η = (idΓ′ × α)φψ. A.M.S. classification number: 51M35.

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,...