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The spectrum $\omega(G)$ of a finite group $G$ is the set of
element orders of $G$. Let $L$ be the projective special linear
group $L_{n}(2)$ with $n\geq3$. First, for all $n\geq3$ we
establish that every finite group $G$ with $\omega(G)=\omega(L)$
has a unique non-abelian composition factor and this factor is
isomorphic to $L$. Second, for some special series of integers $n$
we prove that $L$ is recognizable by spectrum, i.e. every finite
group $G$ with $\omega(G)=\omega(L)$ is isomorphic to $L$.
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