Startseite M, B and Co1 are recognisable by their prime graphs
Artikel
Lizenziert
Nicht lizenziert Erfordert eine Authentifizierung

M, B and Co1 are recognisable by their prime graphs

  • Melissa Lee EMAIL logo und Tomasz Popiel
Veröffentlicht/Copyright: 28. Juli 2022
Journal of Group Theory
Aus der Zeitschrift Journal of Group Theory Band 26 Heft 1

Abstract

The prime graph, or Gruenberg–Kegel graph, of a finite group 𝐺 is the graph Γ ⁢ ( G ) whose vertices are the prime divisors of | G | and whose edges are the pairs { p , q } for which 𝐺 contains an element of order p ⁢ q . A finite group 𝐺 is recognisable by its prime graph if every finite group 𝐻 with Γ ⁢ ( H ) = Γ ⁢ ( G ) is isomorphic to 𝐺. By a result of Cameron and Maslova, every such group must be almost simple, so one natural case to investigate is that in which 𝐺 is one of the 26 sporadic simple groups. Existing work of various authors answers the question of recognisability by prime graph for all but three of these groups, namely the Monster, M , the Baby Monster, B , and the first Conway group, Co 1 . We prove that these three groups are recognisable by their prime graphs.

Acknowledgements

This work was initiated in preparation for a research retreat run by the The University of Auckland’s Algebra & Combinatorics research group, on Waiheke Island in July 2021. We thank all of our fellow participants for an enjoyable and productive retreat. Special thanks are due to Jeroen Schillewaert for organising the retreat, and to Eamonn O’Brien and Gabriel Verret for helpful discussions about the questions that we have managed to answer here. We are also grateful to Chris Parker and an anonymous referee for several suggestions that helped to improve the paper.

  1. Communicated by: Andrea Lucchini

References

[1] 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.1996.0125Suche in Google Scholar

[2] T. Breuer, CTblLib—a GAP package, Version 1.3.2, 2021, http://www.math.rwth-aachen.de/~Thomas.Breuer/ctbllib/. Suche in Google Scholar

[3] P. J. Cameron and N. V. Maslova, Criterion of unrecognizability of a finite group by its Gruenberg–Kegel graph, preprint (2021), https://arxiv.org/abs/2012.01482v2. 10.1016/j.jalgebra.2021.12.005Suche in Google Scholar

[4] J. H. Conway, R. T. Curtis, S. P. Norton, R. A. Parker and R. A. Wilson, Atlas of Finite Groups, Oxford University, Oxford, 1985. Suche in Google Scholar

[5] D. Gorenstein, Finite Groups, 2nd ed., Chelsea, New York, 1980. Suche in Google Scholar

[6] M. Hagie, The prime graph of a sporadic simple group, Comm. Algebra 31 (2003), 4405–4424. 10.1081/AGB-120022800Suche in Google Scholar

[7] I. M. Isaacs, Character Theory of Finite Groups, Academic Press, New York, 1976. Suche in Google Scholar

[8] 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), 79–87. 10.21538/0134-4889-2019-25-4-79-87Suche in Google Scholar

[9] 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, Sib. Math. J. 61 (2020), 1087–1092. 10.1134/S0037446620060099Suche in Google Scholar

[10] V. D. Mazurov and W. Shi, A note to the characterization of sporadic simple groups, Algebra Colloq. 5 (1998), 285–288. Suche in Google Scholar

[11] C. E. Praeger and W. Shi, A characterization of some alternating and symmetric groups, Comm. Algebra 22 (1994), 1507–1530. 10.1080/00927879408824920Suche in Google Scholar

[12] R. A. Wilson et al., Atlas of finite group representations – version 3, http://brauer.maths.qmul.ac.uk/Atlas/v3/matrep/Mmax11G0-f3r204B0. Suche in Google Scholar

[13] A. V. Zavarnitsine, Recognition of finite groups by the prime graph, Algebra Logic 45 (2006), 220–231. 10.1007/s10469-006-0020-9Suche in Google Scholar

[14] The GAP Group, GAP—Groups, algorithms, and programming, version 4.11.1, 2021, https://www.gap-system.org. Suche in Google Scholar
Received: 2021-07-27
Revised: 2022-02-14
Published Online: 2022-07-28
Published in Print: 2023-01-01

© 2022 Walter de Gruyter GmbH, Berlin/Boston

Artikel in diesem Heft

  1. Frontmatter
  2. Some simple biset functors
  3. Explicit universal minimal constants for polynomial growth of groups
  4. Shift dynamics of the groups of Fibonacci type
  5. Finitely generated subgroups and Chabauty topology in totally disconnected locally compact groups
  6. The number of set-orbits of a solvable permutation group
  7. On finite 𝜎-tower groups
  8. Orders of inner-diagonal automorphisms of some simple groups of Lie type
  9. The Lie algebra structure of the degree one Hochschild cohomology of the blocks of the sporadic Mathieu groups
  10. M, B and Co1 are recognisable by their prime graphs
https://doi.org/10.1515/jgth-2021-0119

Artikel in diesem Heft

  1. Frontmatter
  2. Some simple biset functors
  3. Explicit universal minimal constants for polynomial growth of groups
  4. Shift dynamics of the groups of Fibonacci type
  5. Finitely generated subgroups and Chabauty topology in totally disconnected locally compact groups
  6. The number of set-orbits of a solvable permutation group
  7. On finite 𝜎-tower groups
  8. Orders of inner-diagonal automorphisms of some simple groups of Lie type
  9. The Lie algebra structure of the degree one Hochschild cohomology of the blocks of the sporadic Mathieu groups
  10. M, B and Co1 are recognisable by their prime graphs
Jetzt anmelden und 20% sparen
Institutioneller Zugang
Wie funktioniert Authentifizierung?
Haben Sie eine Idee, wie wir unsere Website verbessern können?
Bitte schreiben Sie uns.
© 2025 De Gruyter Brill
Heruntergeladen am 27.9.2025 von https://www.degruyterbrill.com/document/doi/10.1515/jgth-2021-0119/html
Button zum nach oben scrollen