Abstract:
Let $G$ be a finite group. The prime graph of $G$ is denoted by $\Gamma(G)$. It is proved in [1] that if $G$ is a finite group such that $\Gamma(G)=\Gamma(B_p(3))$, where $p>3$ is an odd prime, then $G\ge B_p(3)$ or $C_p(3)$. In this paper we prove the main result that if $G$ is a finite group such that $\Gamma(G)=\Gamma(B_n(3))$, where $n\ge6$, then $G$ has a unique nonabelian composition factor isomorphic to $B_n(3)$ or $C_n(3)$. Also if $\Gamma(G)=\Gamma(B_4(3))$, then $G$ has a unique nonabelian composition factor isomorphic to $B_4(3)$, $C_4(3)$, or $^2D_4(3)$. It is proved in [2] that if $p$ is an odd prime, then $B_p(3)$ is recognizable by element orders. We give a corollary of our result, generalize the result of [2], and prove that $B_{2k+1}(3)$ is recognizable by the set of element orders. Also the quasirecognition of $B_{2k}(3)$ by the set of element orders is obtained.
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