## 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(n

• 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

• 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(n

^{7}), 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

^{2}G_{2}(27) and E_{8}(2) are uniquely determined by the isomorphism types of their Gruenberg--Kegel graphs.Original language | English |
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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 | |

Publication status | Published - 1 Oct 2022 |

## Keywords

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