Recognisability of the sporadic groups by the isomorphism types of their prime graphs

Melissa Lee; Tomasz Popiel

doi:10.1515/jgth-2024-0150

Article

Recognisability of the sporadic groups by the isomorphism types of their prime graphs

Melissa Lee and Tomasz Popiel

Published/Copyright: January 23, 2026

Published by

Become an author with De Gruyter Brill

Submit Manuscript Author Information Explore this Subject

From the journal Journal of Group Theory

Abstract

The prime graph, also called the Gruenberg–Kegel graph, of a finite group 𝐺 is the labelled graph Γ ⁢ ( G ) with vertices the prime divisors of | G | and edges the pairs { p , q } for which 𝐺 contains an element of order p ⁢ q . A group 𝐺 is recognisable by its prime graph if every group 𝐻 with Γ ⁢ ( H ) = Γ ⁢ ( G ) is isomorphic to 𝐺. Cameron and Maslova have shown that every group that is recognisable by its prime graph is almost simple. This justifies the significant amount of attention that has been given to determining which simple or almost simple groups are recognisable by their prime graphs. This problem has been completely solved for certain families of simple groups, including the sporadic groups. A natural extension of the problem is to determine which groups are recognisable by their unlabelled prime graphs, i.e. by the isomorphism types of their prime graphs. Here we determine which of the sporadic finite simple groups are recognisable by the isomorphism types of their prime graphs. We also show that, for every sporadic group 𝐺 that is not recognisable by the isomorphism type of Γ ⁢ ( G ) , there are infinitely many groups 𝐻 with Γ ⁢ ( H ) ≅ Γ ⁢ ( G ) .

Dedicated to Cheryl Praeger on the occasion of her 75th birthday

Funding source: Australian Research Council

Award Identifier / Grant number: DE230100579

Funding statement: The first author acknowledges the support of an Australian Research Council Discovery Early Career Researcher Award, project no. DE230100579.

Acknowledgements

We thank the referees for several helpful comments.

Communicated by: Andrea Lucchini

References

[1] O. A. Alekseeva and A. S. Kondrat’ev, On recognizability of some finite simple orthogonal groups by spectrum, Proc. Steklov Inst. Math. 266 (2009), 10–23. 10.1134/S0081543809060029Search in Google Scholar

[2] A. S. Bang, Talteoretiske undersøgelser, Tidsskrift Math. 4 (1886), 70–80. Search in Google Scholar

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

[4] J. N. Bray, D. F. Holt and C. M. Roney-Dougal, The Maximal Subgroups of the Low-Dimensional Finite Classical Groups. With a Foreword by Martin Liebeck, London Math. Soc. Lecture Note Ser. 407, Cambridge University, Cambridge, 2013. 10.1017/CBO9781139192576Search in Google Scholar

[5] T. Breuer, CTblLib – A GAP package, Version 1.3.4, (2022), http://www.math.rwth-aachen.de/~Thomas.Breuer/ctbllib/. Search in Google Scholar

[6] Y. Bugeaud, Z. Cao and M. Mignotte, On simple K 4 -groups, J. Algebra 241 (2001), no. 2, 658–668. 10.1006/jabr.2000.8742Search in Google Scholar

[7] T. C. Burness and M. Giudici, Classical Groups, Derangements and Primes, Austral. Math. Soc. Lect. Ser. 25, Cambridge University, Cambridge, 2016. 10.1017/CBO9781139059060Search in Google Scholar

[8] P. J. Cameron and N. V. Maslova, Criterion of unrecognizability of a finite group by its Gruenberg–Kegel graph, J. Algebra 607 (2022), 186–213. 10.1016/j.jalgebra.2021.12.005Search in Google Scholar

[9] M.-Z. Chen, N. V. Maslova and M. R. Zinov’eva, On characterization of groups by isomorphism type of Gruenberg–Kegel graph, preprint (2025), https://arxiv.org/pdf/2504.14703. Search in Google Scholar

[10] J. H. Conway, R. T. Curtis, S. P. Norton, R. A. Parker and R. A. Wilson, ATLAS of Finite Groups, Oxford University, Eynsham, 1985. Search in Google Scholar

[11] D. A. Craven, The maximal subgroups of the exceptional groups F 4 ⁢ ( q ) , E 6 ⁢ ( q ) and E 6 2 ⁢ ( q ) and related almost simple groups, Invent. Math. 234 (2023), no. 2, 637–719. 10.1007/s00222-023-01208-2Search in Google Scholar

[12] S. Dolfi, E. Jabara and M. S. Lucido, C ⁢ 55 -groups, Siberian Math. J. 45 (2004), no. 6, 1053–1062. 10.1023/B:SIMJ.0000048920.62281.61Search in Google Scholar

[13] W. Feit and J. G. Thompson, Solvability of groups of odd order, Pacific J. Math. 13 (1963), no. 3, 775–1029. 10.2140/pjm.1963.13.1029Search in Google Scholar

[14] G. C. Gerono, Note sur la résolution en nombres entiers et positifs de l’équation x m = y n + 1 , Nouv. Ann. Math. 9 (1870), 469–471. Search in Google Scholar

[15] M. Giudici, Maximal subgroups of almost simple groups with socle PSL ⁢ ( 2 , q ) , preprint (2007), https://arxiv.org/pdf/math/0703685.pdf. Search in Google Scholar

[16] M. Hagie, The prime graph of a sporadic simple group, Comm. Algebra 31 (2003), no. 9, 4405–4424. 10.1081/AGB-120022800Search in Google Scholar

[17] A. S. Kondrat’ev, On the recognizability of sporadic simple groups R ⁢ u , H ⁢ N , F ⁢ i 22 , H ⁢ e , M ⁢ c ⁢ L and C ⁢ o 3 by the Gruenberg–Kegel graph (in Russian), Trudy Inst. Mat. i Mekh. UrO RAN 25 (2019), no. 4, 79–87. 10.21538/0134-4889-2019-25-4-79-87Search in Google Scholar

[18] A. S. Kondrat’ev, On recognition of the sporadic simple groups H ⁢ S , J 3 , S ⁢ u ⁢ z , O ′ ⁢ N , L ⁢ y , T ⁢ h , F ⁢ i 23 , and F ⁢ i 24 ′ by the Gruenberg–Kegel graph, Siberian Math. J. 61 (2020), no. 6, 1087–1092. 10.1134/S0037446620060099Search in Google Scholar

[19] M. Lee and T. Popiel, M, B and Co 1 are recognisable by their prime graphs, J. Group Theory 26 (2023), no. 1, 193–205. Search in Google Scholar

[20] R. P. Martineau, On 2-modular representations of the Suzuki groups, Amer. J. Math. 94 (1972), no. 1, 55–72. 10.2307/2373593Search in Google Scholar

[21] R. P. Martineau, On representations of the Suzuki groups over fields of odd characteristic, J. Lond. Math. Soc. (2) 6 (1972), 153–160. 10.1112/jlms/s2-6.1.153Search in Google Scholar

[22] N. V. Maslova, V. V. Panshin and A. M. Staroletov, On characterization by Gruenberg–Kegel graph of finite simple exceptional groups of Lie type, Eur. J. Math. 9 (2023), no. 3, Paper No. 78. 10.1007/s40879-023-00672-7Search in Google Scholar

[23] V. D. Mazurov and W. Shi, A note to the characterization of sporadic simple groups, Algebra Colloq. 5 (1998), no. 3, 285–288. Search in Google Scholar

[24] J. Omielan, Answer to a question on Mathematics Stack Exchange, 19 August (2023), https://math.stackexchange.com/q/4761274. Search in Google Scholar

[25] C. E. Praeger and W. J. Shi, A characterization of some alternating and symmetric groups, Comm. Algebra 22 (1994), no. 5, 1507–1530. 10.1080/00927879408824920Search in Google Scholar

[26] M. Suzuki, On a class of doubly transitive groups, Ann. of Math. (2) 75 (1962), no. 1, 105–145. 10.2307/1970423Search in Google Scholar

[27] A. V. Vasil’ev, On connection between the structure of a finite group and the properties of its graph, Siberian Math. J. 46 (2005), no. 3, 396–404. 10.1007/s11202-005-0042-xSearch in Google Scholar

[28] A. V. Vasil’ev and I. B. Gorshkov, On recognition of finite simple groups with connected prime graph, Sib. Math. J. 50 (2009), no. 2, 233–238. 10.1007/s11202-009-0027-2Search in Google Scholar

[29] A. V. Vasil’ev and E. P. Vdovin, An adjacency criterion for the prime graph of a finite simple group, Algebra Logic 44 (2005), no. 6, 381–406. 10.1007/s10469-005-0037-5Search in Google Scholar

[30] A. V. Vasil’ev and E. P. Vdovin, Cocliques of maximal size in the prime graph of a finite simple group, Algebra Logic 50 (2011), no. 4, 291–322. 10.1007/s10469-011-9143-8Search in Google Scholar

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

[32] R. A. Wilson, The Maximal Subgroups of the Lyons Group, Math. Proc. Cambridge Philos. Soc. 97 (1985), no. 3, 433–436. 10.1017/S0305004100063003Search in Google Scholar

[33] A. V. Zavarnitsin, Recognition of finite groups by the prime graph, Algebra Logic 45 (2006), no. 4, 220–231. 10.1007/s10469-006-0020-9Search in Google Scholar

[34] M. R. Zinov’eva and V. D. Mazurov, On finite groups with disconnected prime graph, Proc. Steklov Inst. Math. 283 (2013), no. 1, 139–145. 10.1134/S0081543813090149Search in Google Scholar

[35] K. Zsigmondy, Zur Theorie der Potenzreste, Monatsh. Math. Phys. 3 (1892), no. 1, 265–284. 10.1007/BF01692444Search in Google Scholar

[36] The GAP Group, GAP – Groups, algorithms, and programming, version 4.12.1, 2022, https://www.gap-system.org. Search in Google Scholar

Received: 2024-07-10

Revised: 2025-10-09

Published Online: 2026-01-23

You are currently not able to access this content.

https://doi.org/10.1515/jgth-2024-0150