Home On the strong connectivity of the 2-Engel graphs of almost simple groups
Article
Licensed
Unlicensed Requires Authentication

On the strong connectivity of the 2-Engel graphs of almost simple groups

  • Francesca Dalla Volta EMAIL logo , Fabio Mastrogiacomo and Pablo Spiga ORCID logo
Published/Copyright: September 19, 2023

Abstract

The Engel graph of a finite group ๐บ is a directed graph encoding the pairs of elements in ๐บ satisfying some Engel word. Recent work of Lucchini and the third author shows that, except for a few well-understood cases, the Engel graphs of almost simple groups are strongly connected. In this paper, we give a refinement to this analysis.

Acknowledgements

The authors are members of the GNSAGA INdAM research group and kindly acknowledge their support.

  1. Communicated by: Andrea Lucchini

References

[1] M. Aschbacher, A condition for the existence of a strongly embedded subgroup, Proc. Amer. Math. Soc. 38 (1973), 509โ€“511. 10.1090/S0002-9939-1973-0318308-0Search in Google Scholar

[2] W. Bosma, J. Cannon and C. Playoust, The Magma algebra system. I. The user language, J. Symbolic Comput. 24 (1997), 235โ€“265. 10.1006/jsco.1996.0125Search in Google Scholar

[3] J.โ€‰N. Bray, D.โ€‰F. Holt and C.โ€‰M. Roney-Dougal, The Maximal Subgroups of the Low-Dimensional Finite Classical Groups, London Math. Soc. Lecture Note Ser. 407, Cambridge University, Cambridge, 2013. 10.1017/CBO9781139192576Search in Google Scholar

[4] P.โ€‰J. Cameron, Graphs defined on groups, Int. J. Group Theory 11 (2022), no. 2, 53โ€“107. Search in Google Scholar

[5] J.โ€‰H. Conway, R.โ€‰T. Curtis, S.โ€‰P. Norton, R.โ€‰A. Parker and R.โ€‰A. Wilson, AโขTโขLโขAโขS of Finite Groups, Oxford University, Eynsham, 1985. Search in Google Scholar

[6] E. Detomi, A. Lucchini and D. Nemmi, The Engel graph of a finite group, Forum Math. 35 (2023), no. 1, 111โ€“122. 10.1515/forum-2022-0070Search in Google Scholar

[7] I.โ€‰M. Isaacs, Character Theory of Finite Groups, Pure Appl. Math. 69, Academic Press, New York, 1976. Search in Google Scholar

[8] A.โ€‰S. Kondratโ€™ev and V.โ€‰D. Mazurov, Recognition of alternating groups of prime degree from the orders of their elements, Sib. Math. J. 41 (2000), no. 2, 294โ€“302. 10.1007/BF02674599Search in Google Scholar

[9] M.โ€‰W. Liebeck, J. Saxl and G.โ€‰M. Seitz, Subgroups of maximal rank in finite exceptional groups of Lie type, Proc. Lond. Math. Soc. (3) 65 (1992), no. 2, 297โ€“325. 10.1112/plms/s3-65.2.297Search in Google Scholar

[10] A. Lucchini and P. Spiga, The Engel graph of almost simple groups, to, preprint (2022), https://arxiv.org/abs/2205.14984. Search in Google Scholar

[11] G.โ€‰L. Morgan and C.โ€‰W. Parker, The diameter of the commuting graph of a finite group with trivial centre, J. Algebra 393 (2013), 41โ€“59. 10.1016/j.jalgebra.2013.06.031Search in Google Scholar

[12] M. Muzychuk and P. Spiga, Finite primitive groups of small rank: symmetric and sporadic groups, J. Algebraic Combin. 52 (2020), no. 2, 103โ€“136. 10.1007/s10801-019-00896-5Search in Google Scholar

[13] S.โ€‰P. Norton, Anatomy of the Monster. I, The Atlas of Finite Groups: Ten Years on (Birmingham 1995), London Math. Soc. Lecture Note Ser. 249, Cambridge University, Cambridge (1998), 198โ€“214. 10.1017/CBO9780511565830.020Search in Google Scholar

[14] J.โ€‰S. Williams, Prime graph components of finite groups, J. Algebra 69 (1981), no. 2, 487โ€“513. 10.1016/0021-8693(81)90218-0Search in Google Scholar

Received: 2023-04-19
Revised: 2023-07-31
Published Online: 2023-09-19
Published in Print: 2024-03-01

ยฉ 2023 Walter de Gruyter GmbH, Berlin/Boston

Downloaded on 8.9.2025 from https://www.degruyterbrill.com/document/doi/10.1515/jgth-2023-0060/html
Scroll to top button