8th Croatian Mathematical Congress

The existence of Steiner 2-designs $S(2,6,91)$ having $\textrm{Frob}_{26}$ as an automorphism group

3 Jul 2024, 16:25

20m

D3 (School of Applied Mathematics and Informatics, J. J. Strossmayer University of Osijek)

Talk CDM: Combinatorics and Discrete Mathematics Combinatorics and Discrete Mathematics

Speaker

Doris Dumičić Danilović (Faculty of Mathematics, University of Rijeka, Croatia)

Description

So far, there are only four known Steiner 2-designs $S(2,6,91)$ which have been found by C.J.Colbourn, M.J.Colbourn and W.H.Mills. Each of them is cyclic, i.e. having a cyclic automorphism group acting transitively on points. For more than 30 years no results about that designs have been published, and the last one is from 1991, when Z.Janko and V.D.Tonchev showed that any point-transitive $2$--$(91,6,1)$ design with an automorphism group of order larger than $91$ is one of the four known designs.
In this talk, we show that there are exactly two Steiner $2$--designs $S(2,6,91)$ with a non-abelian automorphism group of order 26 (i.e. the Frobenius group $\textrm{Frob}_{26}$), and they are isomorphic to the already known designs. Still remains an open question whether there exists a $2$--$(91,6,1)$ design which is not cyclic.