Quasiregular collineation groups of finite projective planes

On finite projective planes in Lenz-Barlotti class at least I.3

“…An up-to-date account of the DembowskiPiper classification [6] is given by the present authors [7]. Background on difference sets and group rings can be found in Chapter VI of Beth, Jungnickel and Lenz [2].…”

“…• planes admitting a quasiregular collineation group of type (g) in the DembowskiPiper classification [6];…”

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“…So a large quasiregular group of collineations has exactly one orbit of size |G| on points. One has: Theorem 1.1 (Dembowski-Piper, [DP67], [Dem68, 4.2.10, p.182]). Let G be a quasiregular group of collineations of the projective plane P of order n. Denote by m = m(G) the number of point (or line) orbits of G, and by F = F(G) the substructure of the elements fixed by G. If |G| > 1 2 (n 2 + n + 1), then there are only the following possibilities: DP a |G| = n 2 + n + 1, m = 1 and F = ∅.…”

Proof of the prime power conjecture for projective planes of order 𝑛 with abelian collineation groups of order 𝑛²

“…Let us consider some background. Among other things, Dembowski and Piper [3] showed that there are only three possible types of projective planes of order n with abelian collineation groups G of order n 2 . These are translation planes, dual translation planes and the so-called type (b) planes.…”

“…By a classical result of André [1], in the case of translation planes and dual translation planes, the collineation group G is always an elementary abelian p-group. Following [3], a projective plane of order n is called a type (b) plane if it has an abelian collineation group of order n 2 whose orbits on the point set P are {p}, L \ {p} and P \ L where (p, L) is a suitable incident point-line pair. In this case, we call G a group of type (b).…”