Abstract
The Engel graph of a finite group ๐บ is a directed graph encoding the pairs of elements in ๐บ 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.
Communicated by: Andrea Lucchini
References
[1] M. Aschbacher, A condition for the existence of a strongly embedded subgroup, Proc. Amer. Math. Soc. 38 (1973), 509โ511. 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), 235โ265. 10.1006/jsco.1996.0125Search in Google Scholar
[3] J.โN. Bray, D.โF. Holt and C.โM. 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.โJ. Cameron, Graphs defined on groups, Int. J. Group Theory 11 (2022), no. 2, 53โ107. Search in Google Scholar
[5]
J.โH. Conway, R.โT. Curtis, S.โP. Norton, R.โA. Parker and R.โA. Wilson,
[6] E. Detomi, A. Lucchini and D. Nemmi, The Engel graph of a finite group, Forum Math. 35 (2023), no. 1, 111โ122. 10.1515/forum-2022-0070Search in Google Scholar
[7] I.โM. Isaacs, Character Theory of Finite Groups, Pure Appl. Math. 69, Academic Press, New York, 1976. Search in Google Scholar
[8] A.โS. Kondratโev and V.โD. Mazurov, Recognition of alternating groups of prime degree from the orders of their elements, Sib. Math. J. 41 (2000), no. 2, 294โ302. 10.1007/BF02674599Search in Google Scholar
[9] M.โW. Liebeck, J. Saxl and G.โM. Seitz, Subgroups of maximal rank in finite exceptional groups of Lie type, Proc. Lond. Math. Soc. (3) 65 (1992), no. 2, 297โ325. 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.โL. Morgan and C.โW. Parker, The diameter of the commuting graph of a finite group with trivial centre, J. Algebra 393 (2013), 41โ59. 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, 103โ136. 10.1007/s10801-019-00896-5Search in Google Scholar
[13] S.โP. 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), 198โ214. 10.1017/CBO9780511565830.020Search in Google Scholar
[14] J.โS. Williams, Prime graph components of finite groups, J. Algebra 69 (1981), no. 2, 487โ513. 10.1016/0021-8693(81)90218-0Search in Google Scholar
ยฉ 2023 Walter de Gruyter GmbH, Berlin/Boston
Articles in the same Issue
- Frontmatter
- Presentations of Schur covers of braid groups
- Graphical complexes of groups
- Narrow normal subgroups of Coxeter groups and of automorphism groups of Coxeter groups
- A closure operator on the subgroup lattice of GL(๐,๐) and PGL(๐,๐) in relation to the zeros of the Mรถbius function
- On the strong connectivity of the 2-Engel graphs of almost simple groups
- Finitely generated metabelian groups arising from integer polynomials
- Finite ๐-groups of class two with a small multiple holomorph
- Hall classes in linear groups
- Projective representations of Heisenberg groups over the rings of order ๐2
Articles in the same Issue
- Frontmatter
- Presentations of Schur covers of braid groups
- Graphical complexes of groups
- Narrow normal subgroups of Coxeter groups and of automorphism groups of Coxeter groups
- A closure operator on the subgroup lattice of GL(๐,๐) and PGL(๐,๐) in relation to the zeros of the Mรถbius function
- On the strong connectivity of the 2-Engel graphs of almost simple groups
- Finitely generated metabelian groups arising from integer polynomials
- Finite ๐-groups of class two with a small multiple holomorph
- Hall classes in linear groups
- Projective representations of Heisenberg groups over the rings of order ๐2