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M, B and Co1 are recognisable by their prime graphs

  • Melissa Lee EMAIL logo and Tomasz Popiel
Published/Copyright: July 28, 2022
Journal of Group Theory
From the journal Journal of Group Theory Volume 26 Issue 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

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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

Articles in the same Issue

  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

Articles in the same Issue

  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
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