843 research outputs found
On the Cameron-Praeger Conjecture
This paper takes a significant step towards confirming a long-standing and
far-reaching conjecture of Peter J. Cameron and Cheryl E. Praeger. They
conjectured in 1993 that there are no non-trivial block-transitive 6-designs.
We prove that the Cameron-Praeger conjecture is true for the important case of
non-trivial Steiner 6-designs, i.e. for 6- designs with
, except possibly when the group is P\GammaL(2,p^e) with or
3, and is an odd prime power.Comment: 11 pages; to appear in: "Journal of Combinatorial Theory, Series A
On the existence of block-transitive combinatorial designs
Block-transitive Steiner -designs form a central part of the study of
highly symmetric combinatorial configurations at the interface of several
disciplines, including group theory, geometry, combinatorics, coding and
information theory, and cryptography. The main result of the paper settles an
important open question: There exist no non-trivial examples with (or
larger). The proof is based on the classification of the finite 3-homogeneous
permutation groups, itself relying on the finite simple group classification.Comment: 9 pages; to appear in "Discrete Mathematics and Theoretical Computer
Science (DMTCS)
Divisibility graph for symmetric and alternating groups
Let be a non-empty set of positive integers and .
The divisibility graph has as the vertex set and there is an edge
connecting and with whenever divides or
divides . Let be the set of conjugacy class sizes of a group .
In this case, we denote by . In this paper we will find the
number of connected components of where is the symmetric group
or is the alternating group
- β¦