On finite simple classical groups over fields of different characteristics with coinciding prime graphs

M. R. Zinov'eva

Trudy Instituta matematiki i mehaniki, Trudy Instituta Matematiki i Mekhaniki UrO RAN, Tome 22 (2016) no. 3, pp. 101-116

Résumé

Suppose that $G$ is a finite group, $\pi(G)$ is the set of prime divisors of its order, and $\omega(G)$ is the set of orders of its elements. We define a graph on $\pi(G)$ with the following adjacency relation: different vertices $r$ and $s$ from $\pi(G)$ are adjacent if and only if $rs\in \omega(G)$. This graph is called the $\it{Gruenberg-Kegel\, graph }$ for the $\it{prime\, graph }$ of $G$ and is denoted by $GK(G)$. Let $G$ and $G_1$ be two nonisomorphic finite simple groups of Lie type over fields of orders $q$ and $q_1$, respectively, with different characteristics. It is proved that, if $G$ is a classical group of a sufficiently high Lie rank, then the prime graphs of the groups $G$ and $G_1$ may coincide only in one of three cases. It is also proved that, if $G=A_1(q)$ and $G_1$ is a classical group, then the prime graphs of the groups $G$ and $G_1$ coincide only if $\{G,G_1\}$ is equal to $\{A_1(9),A_1(4)\}$, $\{A_1(9),A_1(5)\}$, $\{A_1(7),A_1(8)\}$, or $\{A_1(49),^2A_3(3)\}$.

Détail

Keywords: finite simple classical group, prime graph, spectrum.