Rad HAZU, Matematičke znanosti, Vol. 29 (2025), 37-61.

THE NON-ABELIAN GROUP OF ORDER 26 ACTING ON STEINER 2–DESIGNS S(2, 6, 91)

Dean Crnković and Doris Dumičić Danilović

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.

2020 Mathematics Subject Classification.   05B05, 05E18.

Key words and phrases.   Steiner system, automorphism group, Frobenius group.

DOI: https://doi.org/10.21857/mjrl3uo289

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