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

Melissa Lee; Tomasz Popiel

doi:10.1515/jgth-2024-0150

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

Melissa Lee und Tomasz Popiel

Veröffentlicht/Copyright: 23. Januar 2026

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Aus der Zeitschrift 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

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Received: 2024-07-10

Revised: 2025-10-09

Published Online: 2026-01-23

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