Simple Groups Which are $2$-Fold OD-Characterizable

M. Akbari ; A. R. Moghaddamfar

Bulletin of the Malaysian Mathematical Society, Tome 35 (2012) no. 1

Résumé

Let $G$ be a finite group and ${\rm D}(G)$ be the degree pattern of $G$. Denote by $h_{{\rm OD}}(G)$ the number of isomorphism classes of finite groups $H$ satisfying $(|H|, {\rm D}(H))=(|G|, {\rm D}(G))$. A finite group $G$ is called $k$- fold {OD}- characterizable if $h_{\rm OD}(G)=k$. As the main results of this paper, we prove that each of the following pairs $\{G_1, \ G_2\}$ of groups:

$ \{B_n(q),\ C_n(q)\},\quad n=2^m> 2, \quad |\pi\left(\frac{q^n+1}{2}\right)|=1, \quad q\; \text{is odd prime \ power};\\ \{B_p(3), \ C_p(3)\}, \quad |\pi\left(\frac{3^p-1}{2}\right)|=1, \quad p\; \text{ is an odd prime,}\\ \{B_3(5), \ C_3(5)\}, $

satisfies $h_{{\rm OD}}(G_i)=2$, $i=1, 2$. We also prove that, if $(1)$ $n=2$ and $q$ is any prime power such that $|\pi({q^2+1}/{(2, q-1)})|=1$ or $(2)$ $n=2^m\geq 2$ and $q$ is a power of 2 such that $|\pi(q^n+1)|=1$, then $h_{{\rm OD}}(C_n(q))=h_{{\rm OD}}(B_n(q))=1$.

Détail

Classification : 20D05, 20D06, 20D08.