Criterion of unrecognizability of a finite group by its Gruenberg–Kegel graph

Peter J. Cameron; Natalia Maslova

doi:10.1016/j.jalgebra.2021.12.005

Criterion of unrecognizability of a finite group by its Gruenberg–Kegel graph

Peter J. Cameron, Natalia Maslova^*

^*Corresponding author for this work

Research output: Contribution to journal › Article › peer-review

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.

Original language	English
Pages (from-to)	186-213
Number of pages	28
Journal	Journal of Algebra
Volume	607
Issue number	Part A
Early online date	14 Dec 2021
DOIs	https://doi.org/10.1016/j.jalgebra.2021.12.005
Publication status	Published - 1 Oct 2022

Keywords

Finite group
Gruenberg-Kegel graph (prime graph)
Almost simple group
Recognition by Gruenberg-Kegel graph (prime graph)

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Cameron-2021-Criterion-of-unrecognizability-JAlgebra-AAM
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Cite this

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title = "Criterion of unrecognizability of a finite group by its Gruenberg–Kegel graph",

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.",

keywords = "Finite group, Gruenberg-Kegel graph (prime graph), Almost simple group, Recognition by Gruenberg-Kegel graph (prime graph)",

author = "Cameron, {Peter J.} and Natalia Maslova",

note = "Funding: This work was supported by the Russian Science Foundation (project no. 19-71-10067).",

year = "2022",

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doi = "10.1016/j.jalgebra.2021.12.005",

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T1 - Criterion of unrecognizability of a finite group by its Gruenberg–Kegel graph

AU - Cameron, Peter J.

AU - Maslova, Natalia

N1 - Funding: This work was supported by the Russian Science Foundation (project no. 19-71-10067).

PY - 2022/10/1

Y1 - 2022/10/1

N2 - 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.

AB - 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.

KW - Finite group

KW - Gruenberg-Kegel graph (prime graph)

KW - Almost simple group

KW - Recognition by Gruenberg-Kegel graph (prime graph)

U2 - 10.1016/j.jalgebra.2021.12.005

DO - 10.1016/j.jalgebra.2021.12.005

M3 - Article

SN - 0021-8693

VL - 607

SP - 186

EP - 213

JO - Journal of Algebra

JF - Journal of Algebra

IS - Part A

ER -

Criterion of unrecognizability of a finite group by its Gruenberg–Kegel graph

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Keywords

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