Sugli insiemi di permutazioni strettamente k-transitivi (k ? 3)

Abstract

LUIGI ANTONIO ROSATI SUGLI INSIEMI DI PERMUTAZIONI STRETTAMENTE k-TRANSITIVI (k>/3)* Dedicato a Guido Zappa in occasione del suo 70 ° compleannoSUMMARY. Let G be a sharply 3-transitive permutation set on a finite set E of even cardinality and let 1 be in G. The following theorems are proved.

Ovals in a plane coordinatised by a regular nearfield of dimension 2 over its centre

“…In [7], Rosati gave a nice description of π using permutation group theory. Let G be a sharply 3-transitive permutation group on a set E of degree n + 1.…”

“…Clearly, a,b,c,d has size n − 1. The following result is due to Rosati[7]. LEMMA 2.10. a,b,c,d is a hyperbolic oval whose infinite points are X ∞ and Y ∞ .…”

“…In [7] Rosati showed that every sharply 3-transitive permutation group G gives rise to an affine plane π whose projective closure contains ovals. Such ovals are of hyperbolic type as they have two infinite points.…”

Over the non-existence of sharply 3-transitive permutation sets containing sharply 2-transitive permutation subsets

“…Each circle of M not through p=(a,b)~i.e, each cceG such that c~(a) ¢ b -determines an oval of Mp, the points of which are the ideal points of the two classes of generators and the points (x, c~(x)) with xeE\{a, c~-l(b)} (see [17], [19]). By an oval in a finite projective plane of order n we mean each (n + 1)-arc, i.e.…”