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Volume 35 • Number 1 • 2012
• Simple Groups Which are $2$-Fold OD-Characterizable
M. Akbari and A. R. Moghaddamfar

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: \begin{align*} &\{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)\}, \end{align*} 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 $h_{{\rm OD}}(C_n(q))=h_{{\rm OD}}(B_n(q))=1$.

2010 Mathematics Subject Classification: 20D05, 20D06, 20D08.

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