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Criterion of unrecognizability of a finite group by its Gruenberg–Kegel graph
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dc.contributor.author | Cameron, Peter J. | |
dc.contributor.author | Maslova, Natalia | |
dc.date.accessioned | 2022-12-14T00:40:28Z | |
dc.date.available | 2022-12-14T00:40:28Z | |
dc.date.issued | 2022-10-01 | |
dc.identifier | 277014574 | |
dc.identifier | ce4aee67-8e5d-4bc8-89df-c872c1272281 | |
dc.identifier | 85121564751 | |
dc.identifier | 000833526600008 | |
dc.identifier.citation | Cameron , P J & Maslova , N 2022 , ' Criterion of unrecognizability of a finite group by its Gruenberg–Kegel graph ' , Journal of Algebra , vol. 607 , no. Part A , pp. 186-213 . https://doi.org/10.1016/j.jalgebra.2021.12.005 | en |
dc.identifier.issn | 0021-8693 | |
dc.identifier.other | ORCID: /0000-0003-3130-9505/work/105318436 | |
dc.identifier.uri | https://hdl.handle.net/10023/26585 | |
dc.description | Funding: This work was supported by the Russian Science Foundation (project no. 19-71-10067). | en |
dc.description.abstract | The Gruenberg--Kegel graph Γ(G) associated with a finite group G is an undirected graph without loops and multiple edges whose vertices are the prime divisors of |G| and in which vertices p and q are adjacent in Γ(G) if and only if G contains an element of order pq. This graph has been the subject of much recent interest; one of our goals here is to give a survey of some of this material, relating to groups with the same Gruenberg--Kegel graph. However, our main aim is to prove several new results. Among them are the following. • There are infinitely many finite groups with the same Gruenberg--Kegel graph as the Gruenberg--Kegel of a finite group G if and only if there is a finite group H with non-trivial solvable radical such that Γ(G)=Γ(H). • There is a function F on the natural numbers with the property that if a finite n-vertex graph whose vertices are labelled by pairwise distinct primes is the Gruenberg--Kegel graph of more than F(n) finite groups, then it is the Gruenberg--Kegel graph of infinitely many finite groups. (The function we give satisfies F(n)=O(n7), but this is not best possible.) • If a finite graph Gamma whose vertices are labelled by pairwise distinct primes is the Gruenberg--Kegel graph of only finitely many finite groups, then all such groups are almost simple; moreover, Gamma has at least three pairwise non-adjacent vertices, and each vertex is non-adjacent to at least one other vertex, in particular, 2 is non-adjacent to at least one odd vertex. • Groups whose power graphs, or commuting graphs, are isomorphic have the same Gruenberg--Kegel graph. • The groups 2G2(27) and E8(2) are uniquely determined by the isomorphism types of their Gruenberg--Kegel graphs. | |
dc.format.extent | 28 | |
dc.format.extent | 477926 | |
dc.language.iso | eng | |
dc.relation.ispartof | Journal of Algebra | en |
dc.subject | Finite group | en |
dc.subject | Gruenberg-Kegel graph (prime graph) | en |
dc.subject | Almost simple group | en |
dc.subject | Recognition by Gruenberg-Kegel graph (prime graph) | en |
dc.subject | QA Mathematics | en |
dc.subject | T-NDAS | en |
dc.subject.lcc | QA | en |
dc.title | Criterion of unrecognizability of a finite group by its Gruenberg–Kegel graph | en |
dc.type | Journal article | en |
dc.contributor.institution | University of St Andrews. Pure Mathematics | en |
dc.contributor.institution | University of St Andrews. Centre for Interdisciplinary Research in Computational Algebra | en |
dc.identifier.doi | https://doi.org/10.1016/j.jalgebra.2021.12.005 | |
dc.description.status | Peer reviewed | en |
dc.date.embargoedUntil | 2022-12-14 |
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