A permutation group $G$ of a finite set $\Omega$ acts componentwisely on the Cartesian square $\Omega^2$. The largest subgroup of Sym$(\Omega)$ having the same orbits on $\Omega^2$ as $G$ is called the $2$-closure of $G$. The rank of $G$ is the number of its orbits on $\Omega^2$. If the rank of $G$ is $3$ and the order is even, then an undirected graph with vertex set $\Omega$ is defined up to taking complement, for which one of the two off-diagonal orbits of $G$ on $\Omega^2$ is taken as the edge set. Such a graph is called a graph of rank $3$. The full automorphism group of this graph coincides with the $2$-closure of $G$ and contains $G$ as a subgroup. At present, except for the case when $G$ is an almost simple group, there is an explicit description of the $2$-closures of groups $G$ of rank $3$. In this paper, we fill the existing gap, thereby completing the description of the complete automorphism groups of graphs of rank $3$.
Keywords: almost simple group, $2$-closure of permutation group, rank $3$ permutation group, rank $3$ graph, the automorphism group of a graph
Received October 12, 2024
Revised December 6, 2024
Accepted December 9, 2024
Published online December 12, 2024
Funding Agency: The research of A.V. Vasil’ev and D.O. Revin was carried out within the State Contract of the Sobolev Institute of Mathematics (FWNF-2022-0002).
Zhigang Wang, PhD, Prof., School of Mathematics and Statistics, Hainan Univ., Haikou, Hainan, 570225, P. R. China, e-mail: wzhigang@hainanu.edu.cn
Andrey Viktorovich Vasil’ev, Dr. Phys.-Math. Sci., Prof., Sobolev Institute of Mathematics of the Siberia Branch of the Russian Academy of Sciences, Novosibirsk, 630090 Russia, e-mail: vasand@math.nsc.ru
Danila Olegovich Revin, Dr. Phys.-Math. Sci., Prof., Sobolev Institute of Mathematics of the Siberia Branch of the Russian Academy of Sciences, Novosibirsk, 630090 Russia; Krasovskii Institute of Mathematics and Mechanics of the Ural Branch of the Russian Academy of Sciences, Yekaterinburg, 620108 Russia, e-mail: revin@math.nsc.ru
REFERENCES
1. Bannai E. Maximal subgroups of low rank of finite symmetric and alternating groups. J. Fac. Sci. Univ. Tokyo Sect. IA Math., 1971/72 , vol. 18, pp. 475–486.
2. Brouwer A.E., Van Maldeghem H. Strongly regular graphs. Ser. Encyclopedia Math. and its Appl., vol. 182, Cambridge: Camb. Univ. Press, 2022, 425 p. https://doi.org/10.1017/9781009057226
3. Brouwer A.E., Cohen A.M., Neumeier A. Distance-regular graphs. Berlin, Heidelberg, NY: Springer-Verlag, 1989, 495 p. https://doi.org/10.1007/978-3-642-74341-2
4. Bray J., Holt D., Roney-Dougal C. The maximal subgroups of the low-dimensional finite classical groups. London Math. Soc. Lect. Note Ser., vol. 407, Cambridge: Cambridge Univ. Press, 2013, 438 p. https://doi.org/10.1017/CBO9781139192576
5. Conway J.H., et al. Atlas of finite groups. Oxford: Clarendon Press, 1985, 252 p.
6. Craven D.A. The maximal subgroups of the exceptional groups $F_4(q)$, $E_6(q)$, and ${}^2E_6(q)$ and related almost simple groups. Invent. Math., 2023, vol. 234, pp. 637–719. https://doi.org/10.1007/s00222-023-01208-2
7. Dixon J.D., Mortimer B. Permutation groups. Ser. Graduate Texts in Mathematics, vol. 163, NY: Springer, 1996, 348 p. https://doi.org/10.1007/978-1-4612-0731-3
8. Gorenstein D., Lyons R., Solomon R. The classification of the finite simple groups, Number 3, Part I, Ch. A: Almost simple $\mathcal{K}$-groups, Ser. Math. Surv. Monogr., 40.3, Providence, RI: Ams. Math. Soc., 1998.
9. Grechkoseeva M.A. On spectra of almost simple extensions of even-dimensional orthogonal groups. Siberian Math. J., 2018, vol. 59, no. 4, pp. 623–640. https://doi.org/10.1134/S0037446618040055
10. Guo J., Vasil’ev A.V., Wang Z. The automorphism groups of small affine rank 3 graphs. 12 p. Available at https://arxiv.org/abs/2410.04341 . https://doi.org/10.48550/arXiv.2410.04341
11. Kantor W.M., Liebler R.A. The rank 3 permutation representations of the finite classical groups. Trans. Amer. Math. Soc., 1982, vol. 271, pp. 1–71. https://doi.org/10.1017/CBO9780511629235
12. Kleidman P., Liebeck M. The subgroup structure of the finite classical groups. Cambridge: University Press, 1990.
13. Liebeck M.W., Praeger C.E., Saxl J. On the 2-closures of finite permutation groups. J. London Math. Soc. (2), 1988, vol. 37., no. 2, pp. 241–252. https://doi.org/10.1112/JLMS/S2-37.2.241
14. Liebeck M.W., Saxl J. The finite primitive permutation groups of rank three. Bull. London Math. Soc., 1986, vol. 18, pp. 165–172. https://doi.org/10.1112/blms/18.2.165
15. Praeger C.E., Saxl J. Closures of finite primitive permutation groups. Bull. London Math. Soc., 1992, vol. 24, pp. 251–258. https://doi.org/10.1112/BLMS/24.3.251
16. Quick M. Probabilistic generation of wreath products of non-abelian finite simple groups. Comm. Algebra., 2004, vol. 32, no. 12, pp. 4753–4768. https://doi.org/10.1142/S0218196706003074
17. Skresanov S.V. On 2-closures of rank 3 groups. Ars Math. Contemp., 2021, vol. 21, no. 1, paper no. 1.08. https://doi.org/10.26493/1855-3974.2450.1dc
18. Steinberg, R. Automorphisms of finite linear groups. Canadian J. Math., 1960, vol. 12, pp. 606–615. https://doi.org/10.4153/CJM-1960-054-6
Cite this article as: Z. Wang, A.V. Vasil’ev, D.O. Revin. On the almost simple automorphism groups of rank 3 graphs. Trudy Instituta Matematiki i Mekhaniki UrO RAN, 2025, vol. 31, no. 1, pp. 36–52.
