Some Quantitative Characterizations of Certain Symplectic Groups

Abstract

Given a finite group $G$, denote by $\mathrm{D}(G)$ the degree pattern of $G$ and by $\mathrm{OC}(G)$ the set of all order components of $G$. Denote by $h_{\mathrm{OD}}(G)$ (resp. $h_{\mathrm{OC}}(G)$) the number of isomorphism classes of finite groups $H$ satisfying conditions $|H|=|G|$ and $\mathrm{D}(H)=\mathrm{D}(G)$ (resp. $\mathrm{OC}(H)=\mathrm{OC}(G)$). A finite group $G$ is called OD-characterizable (resp. OC-characterizable) if $h_{\mathrm{OD}}(G)=1$ (resp. $h_{\mathrm{OC}}(G)=1$). Let $C=C_p(2)$ be a symplectic group over binary field, for which $2^p-1>7$ is a Mersenne prime. The aim of this article is to prove that $h_{\mathrm{OD}}(C)=1=h_{\mathrm{OC}}(C)$.

References in corpus (1)

• OD-Characterization of Some Linear Groups Over Binary Field and Their Automorphism