Finite simple groups of Lie type over a~field of the same characteristic with the same prime graph

M. R. Zinov'eva

Trudy Instituta matematiki i mehaniki, Trudy Instituta Matematiki i Mekhaniki UrO RAN, Tome 20 (2014) no. 2, pp. 168-183

Résumé

For a finite group $G$, let $\pi(G)$ be the set of prime divisors of its order, and let $\omega(G)$ be the set of orders of its elements. Define on $\pi(G)$ a graph with the following adjacency relation: distinct vertices $r$ and $s$ from $\pi(G)$ are adjacent if and only if $rs\in\omega(G)$. This graph is called the Grunberg–Kegel graph or prime graph of the group $G$ and is denoted by $GK(G)$. We prove that, if $G$ and $G_1$ are nonisomorphic finite simple groups of Lie type over fields of orders $q$ and $q_1$, respectively, of the same characteristic, then the graphs $GK(G)$ and $GK(G_1)$ coincide if and only if either $\{G,G_1\}=\{A_1(8),A_2(2)\}$ or $q=q_1$ and the pair $\{G,G_1\}$ coincides with one of the pairs $\{B_n(q),C_n(q)\}$ for odd $q$, $\{B_3(q),D_4(q)\}$, and $\{C_3(q),D_4(q)\}$.

Détail

Keywords: finite simple group of Lie type, prime graph, spectrum.