On the Sylow Normalizers of Some Simple Classical Groups

N. Ahanjideh ; A. Iranmanesh

Bulletin of the Malaysian Mathematical Society, Tome 35 (2012) no. 2

Résumé

Let $G$ be a finite group and $\pi(G)$ be the set of prime divisors of the order of $G$. For $t \in \pi(G)$ denote by $n_t(G)$ the order of a normalizer of $t$-Sylow subgroup of $G$ and put $n(G) = \{n_t(G):t \in \pi(G)\}$. In this paper, we give an answer to the following problem, for the groups of Lie type $B_n$, $C_n$ and $D_n$: "Let L be a finite non-abelian simple group and $G$ be a finite group with $n(L) = n(G)$. Is it true that $L\cong G$?" In this paper, we find the first examples of non-abelian finite simple groups which are not isomorphic and they have the same set of orders of Sylow normalizers and hence, we show that the question above is not correct always. Let $\mathcal{A}$ be the set of prime numbers of order $2n$, $2(n-1)$ and $2(n-2)$ mod $q$. The latter condition is necessary if $n \geq 5$. Also, we show that $D_{n+1}(q)$ is determined uniquely by its order and $\{n_t(D_{n+1}(q)): t \in \mathcal{A} \cup \{2\}\}$ and if $n =2$ or $q \not\equiv \pm 1~({\rm mod}~8) $, then $B_n(q)$ and $C_n(q)$ are characterizable by their orders and orders of $t$-Sylow normalizers, where $t \in \mathcal{A} \cup \{2\}$. If $n \geq 3$ and $q \equiv \pm 1~({\rm mod}~8) $, then $B_n(q)$ and $C_n(q)$ are $2$-characterizable by their orders and the orders of $t$-Sylow normalizers, where $t \in \mathcal{A} \cup \{2\}$.

Détail

Classification : 20D06, 20D20, 20G40, 20C33.