Original scientific paper

https://doi.org/10.21857/mjrl3uo289

The non-abelian group of order 26 acting on Steiner 2-designs S(2, 6, 91)

Dean Crnković ; Faculty of Mathematics, University of Rijeka, Radmile Matejčić 2, 51000 Rijeka, Croatia

Abstract

There are only four known Steiner 2-designs S(2, 6, 91), the Mills design, the McCalla design and two designs found by C. J. Colbourn and M. J. Colbourn. All these designs admit a cyclic automorphism of order 91. In 1991, Z. Janko and V. D. Tonchev showed that any point-transitive Steiner 2-design S(2, 6, 91) with an automorphism group of order larger than 91 is one of the four known designs. It is an open question whether there exists a Steiner 2-design S(2, 6, 91) with full automorphism group of order smaller than 91. In this paper we show that any Steiner 2-design S(2, 6, 91) having a non-abelian automorphism group of order 26 (i.e. the Frobenius group Frob26) is isomorphic to one of the known designs, the McCalla design having the full automorphism group isomorphic to C91 : C12 or the Colbourn and Colbourn design having the full automorphism group isomorphic to C91 : C4.

Keywords

Steiner system; automorphism group, Frobenius group

Hrčak ID:

326513

URI

https://hrcak.srce.hr/326513

Publication date:

9.1.2025.

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