3 research outputs found
Block-transitive 2-designs with a chain of imprimitive partitions
More than years ago, Delandtsheer and Doyen showed that the automorphism
group of a block-transitive -design, with blocks of size , could leave
invariant a nontrivial point-partition, but only if the number of points was
bounded in terms of . Since then examples have been found where there are
two nontrivial point partitions, either forming a chain of partitions, or
forming a grid structure on the point set. We show, by construction of infinite
families of designs, that there is no limit on the length of a chain of
invariant point partitions for a block-transitive -design. We introduce the
notion of an `array' of a set of points which describes how the set interacts
with parts of the various partitions, and we obtain necessary and sufficient
conditions in terms of the `array' of a point set, relative to a partition
chain, for it to be a block of such a design
Linear spaces with a line-transitive point-imprimitive automorphism group and Fang-Li parameter gcd(k,r) at most eight
In 1991, Weidong Fang and Huiling Li proved that there are only finitely many
non-trivial linear spaces that admit a line-transitive, point-imprimitive group
action, for a given value of gcd(k,r), where k is the line size and r is the
number of lines on a point. The aim of this paper is to make that result
effective. We obtain a classification of all linear spaces with this property
having gcd(k,r) at most 8. To achieve this we collect together existing theory,
and prove additional theoretical restrictions of both a combinatorial and group
theoretic nature. These are organised into a series of algorithms that, for
gcd(k,r) up to a given maximum value, return a list of candidate parameter
values and candidate groups. We examine in detail each of the possibilities
returned by these algorithms for gcd(k,r) at most 8, and complete the
classification in this case.Comment: 47 pages Version 1 had bbl file omitted. Apologie