The Gruenberg-Kegel graph $Γ(G)$ associated with a finite group $G$ has as vertices the prime divisors of $|G|$, with an edge from $p$ to $q$ 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 $F(n)=O(n^7)$, but this is probably not best possible.)
- If a finite graph $Γ$ 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, $Γ$ has at least three pairwise non-adjacent vertices, and $2$ 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.
In addition, we consider groups whose Gruenberg-Kegel graph has no edges. These are the groups in which every element has prime power order, and have been studied under the name \emph{EPPO groups}; completing this line of research, we give a complete list of such groups.
On characterization by Gruenberg-Kegel graph of finite simple exceptional groups of Lie type
有限群$G$的Gruenberg-Kegel图$Γ(G)$是其顶点集是$|G|$的素因子集且其中两个不同顶点$r$和$s$相邻的图当且仅当在$G$中存在一个$rs$阶元素。如果只有有限个两两非同构的有限群具有Gruenberg-Kegel图作为$G$，则称有限群$G$几乎可识别（通过Gruenberg-Kegel图）。如果$G$不是几乎可识别的，则称之为不可识别的（根据Gruenberg-Kegel图）。最近，P. J. Cameron和第一作者证明了，如果一个有限群几乎可识别，那么这个群几乎是单的。因此，哪些几乎单群（特别是有限单群）几乎可识别的问题是最重要的。本文证明了：任何既不同构于${^2}B_2(2^{2n+1})$ with $n\geq1$，也不同构于$G_2(3)$，且Gruenberg-Kegel图至少有三个连通分支的有限李型单例外群几乎是可识别的.此外，无法识别组$ {^2}B_2(2^{2n+1})$，其中$n\geq1$和$G_2(3)$。
On the Gruenberg-Kegel Graph of Integral Group Rings of Finite Groups
素图问题询问整群环$\mathbb Z G$的Gruenberg-Kegel图，即$\mathbb Z G$的归一化单位群的素图是否与群$G$的素图重合。本文证明了对于有限群$G$，素图问题约化为几乎单群.我们将这一约化应用于阶至多可被三个素数整除的有限群$G$，并证明了这类群的Gruenberg - Kegel图与$G$的素图重合.
