论文标题
有限群的Gruenberg-Kegel图的标准
Criterion of unrecognizability of a finite group by its Gruenberg-Kegel graph
论文作者
论文摘要
与有限组$ g $相关的gruenberg-kegel图$γ（g）$作为$ | g | $的素数为顶点，从$ p $到$ q $时，只有$ g $包含订单$ pq $的元素。该图一直是最近引起人们关注的主题。我们这里的目标之一是对具有相同Gruenberg-Kegel图的群体进行调查。但是，我们的主要目的是证明几个新结果。其中包括以下内容。 - 有许多有限的团体，具有与有限的组$ g $的Gruenberg-Kegel图相同的Gruenberg-Kegel图，并且仅当有有限的组$ h $带有不可溶解的自由基，以便$γ（g）=γ（h）$。 - 自然数字上有一个函数$ f $，如果有限的$ n $ n $ vertex图，其顶点被成对的不同素数标记为gruenberg-kegel图的gruenberg-kegel图超过$ f（n）$有限群，则它是Gruenberg-Keggel图形无限的许多有限群体的图。 （我们给出的功能满足$ f（n）= o（n^7）$，但这可能不是最好的。） - 如果有限的图形$γ$由成对的素数标记为仅有限的有限群的Gruenberg-Kegel图，则所有此类组几乎都很简单；此外，$γ$至少具有三个成对的非标准顶点，而$ 2 $对至少一个奇数顶点不可种。 - 功率图或通勤图的组为同构具有相同的Gruenberg-Kegel图。 - 组$ {^2} g_2（27）$和$ e_8（2）$由其gruenberg-kegel图的同构类型唯一确定。 此外，我们考虑了Gruenberg-Kegel图没有边缘的组。这些是每个元素具有主要功率顺序的组，并以\ emph {eppo组的名称进行了研究；完成这一研究，我们提供了此类群体的完整列表。
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.
