Glasnik Matematicki, Vol. 56, No. 2 (2021), 225-240. \( \)

GROUPS \(S_{n}\times S_{m}\) IN CONSTRUCTION OF FLAG-TRANSITIVE BLOCK DESIGNS

Snježana Braić, Joško Mandić, Aljoša Šubašić, Tanja Vojković and Tanja Vučičić

Faculty of science, University of Split, 21 000 Split, Croatia
e-mail:sbraic@pmfst.hr
e-mail:majo@pmfst.hr
e-mail:aljsub@pmfst.hr
e-mail:tanja@pmfst.hr
e-mail:vucicic@pmfst.hr

Abstract.   In this paper, we observe the possibility that the group \(S_{n}\times S_{m}\) acts as a flag-transitive automorphism group of a block design with point set \(\{1,\ldots ,n\}\times \{1,\ldots ,m\},4\leq n\leq m\leq 70\). We prove the equivalence of that problem to the existence of an appropriately defined smaller flag-transitive incidence structure. By developing and applying several algorithms for the construction of the latter structure, we manage to solve the existence problem for the desired designs with \(nm\) points in the given range. In the vast majority of the cases with confirmed existence, we obtain all possible structures up to isomorphism.

2020 Mathematics Subject Classification.   05B05, 20B25

Key words and phrases.   Combinatorial designs, incidence structures, automorphism groups

https://doi.org/10.3336/gm.56.2.02

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