Biplanes (56, 11, 2) with involutory collineation fixing 6 points

Intransitive collineation groups of ovals fixing a triangle

We investigate collineation groups of a finite projective plane of odd order fixing an oval and having two orbits on it, one of which is assumed to be primitive. The situation in which there exists a fixed triangle off the oval is considered in detail. Our main result is the following. Theorem. Let p be a finite projective plane of odd order n containing an oval O: If a collineation group G of ...

There are exactly five biplanes with k=11

A biplane is a 2-(k(k − 1)/2 + 1, k, 2) symmetric design. Only sixteen nontrivial biplanes are known: there are exactly nine biplanes with k < 11, at least five biplanes with k = 11, and at least two biplanes with k = 13. It is here shown by exhaustive computer search that the list of five known biplanes with k = 11 is complete. This result further implies that there exists no 3-(57, 12, 2) des...

A Steiner 2-design with an Automorphism Fixing Exactly R + 2 Points

Doyen conjectured that there is no Steiner 2-design having an automorphism with more than r + 1 but fewer than r + p r ? 1 xed points, where r is the replication number. The falsity of this conjecture is shown by describing 2-(45,5,1) designs having an automorphism of order 2 with exactly 13 xed points.

An involutory Pascal matrix

An involutory upper triangular Pascal matrix Un is investigated. Eigenvectors of Un and of UT n are considered. A characterization of Un is obtained, and it is shown how the results can be extended to matrices over a commutative ring with unity. © 2004 Elsevier Inc. All rights reserved.

56 Session 6 Networks of Agents