36 research outputs found
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
Almost simple groups as flag-transitive automorphism groups of 2-designs with {\lambda} = 2
In this article, we study -designs with admitting a
flag-transitive almost simple automorphism group with socle a finite simple
exceptional group of Lie type, and we prove that such a -design does not
exist. In conclusion, we present a classification of -designs with
admitting flag-transitive and point-primitive automorphism groups
of almost simple type, which states that such a -design belongs to an
infinite family of -designs with parameter set and
for some , or it is isomorphic to the -design with
parameter set , , , , ,
, , , or
On flag-transitive 2-(k^2, k, λ) designs with λ | k
It is shown that, apart from the smallest Ree group, a flag-transitive
automorphism group G of a 2-(k^2, k, λ) design D, with λ | k, is either an affine
group or an almost simple classical group. Moreover, when G is the smallest Ree
group, D is isomorphic either to the 2-(6^2, 6, 2) design or to one of the three 2-
(6^2, 6, 6) designs constructed in this paper. All the four 2-designs have the 36
secants of a non-degenerate conic C of PG(2,8) as a point set and 6-sets of secants
in a remarkable configuration as a block set
Sporadic simple groups as flag-transitive automorphism groups of symmetric designs
In this article, we study symmetric designs admitting flag-transitive,
point-imprimitive almost simple automorphism groups with socle sporadic simple
groups. As a corollary, we present a classification of symmetric designs
admitting flag-transitive automorphism group whose socle is a sporadic simple
group, and in conclusion, there are exactly seven such designs, one of which
admits a point-imprimitive automorphism group and the remaining are
point-primitive