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Let $G$ be a finite group. The main result of this paper is as
follows: If $G$ is a finite group, such that $\Gamma(G) =
\Gamma(^{2}G_{2}(q))$, where $q = 3^{2n+1}$ for some $n \geq 1$,
then $G$ has a (unique) nonabelian composition factor isomorphic
to $^{2}G_{2}(q)$. As a consequence, we prove that if $G$ is a
finite group satisfying $|G| = |^{2}G_{2}(q)|$ and $\Gamma(G) =
\Gamma(^{2}G_{2}(q))$ then $G\cong^{2}G_{2}(q)$. This enables us
to give new proofs for some theorems; e.g., a conjecture of W. Shi
and J. Bi. Applications of this result are also considered to the
problem of recognition by element orders of finite groups.
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