182 research outputs found
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
Pairwise transitive 2-designs
We classify the pairwise transitive 2-designs, that is, 2-designs such that a
group of automorphisms is transitive on the following five sets of ordered
pairs: point-pairs, incident point-block pairs, non-incident point-block pairs,
intersecting block-pairs and non-intersecting block-pairs. These 2-designs fall
into two classes: the symmetric ones and the quasisymmetric ones. The symmetric
examples include the symmetric designs from projective geometry, the 11-point
biplane, the Higman-Sims design, and designs of points and quadratic forms on
symplectic spaces. The quasisymmetric examples arise from affine geometry and
the point-line geometry of projective spaces, as well as several sporadic
examples.Comment: 28 pages, updated after review proces
Large dimensional classical groups and linear spaces
Suppose that a group has socle a simple large-rank classical group.
Suppose furthermore that acts transitively on the set of lines of a linear
space . We prove that, provided has dimension at least 25,
then acts transitively on the set of flags of and hence the
action is known. For particular families of classical groups our results hold
for dimension smaller than 25.
The group theoretic methods used to prove the result (described in Section 3)
are robust and general and are likely to have wider application in the study of
almost simple groups acting on finite linear spaces.Comment: 32 pages. Version 2 has a new format that includes less repetition.
It also proves a slightly stronger result; with the addition of our
"Concluding Remarks" section the result holds for dimension at least 2
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