3 research outputs found
Non-transversal Vectors of Some Finite Geometries
By means of associated structural invariants, we efficiently construct four
biplanes of order 9 - except the one with the smallest automorphism group, that
is found by Janko and Trung. The notion of non-transversal vector is introduced
since we observed related properties that provide significantly more efficient
constructions. There is a dichotomy in the structure of biplanes of order 7 and
9 with respect to the incidence matrix symmetry.Comment: 12 page
A class of symmetric association schemes as inclusion of biplanes
Let be a nontrivial biplane of order represented by
symmetric canonical incidence matrix with trace . We proved
that includes a partially balanced incomplete design with
association scheme of three classes. Consequently, these structures are
symmetric, having points. While it is not known whether this class is
finite or infinite, we show that there is a related superclass with infinitely
many representatives.Comment: 11 pages, 2 figure
Symmetries of biplanes
In this paper, we first study biplanes with parameters
, where the block size . These are the smallest
parameter values for which a classification is not available. We show that if
, then either is the Aschbacher biplane or its dual, or
is a subgroup of the cyclic group of order . In the case
where , we prove that divides . We also provide an example of a biplane with
parameters with a flag-transitive and point-primitive subgroup of
automorphisms preserving a homogeneous cartesian decomposition. This motivated
us to study biplanes with point-primitive automorphism groups preserving a
cartesian decomposition. We prove that such an automorphism group is either of
affine type (as in the example), or twisted wreath type.Comment: 24 page