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Let $G$ be a non-abelian group. We define the
noncommuting graph $\nabla(G)$ of $G$ as follows: its vertex set
is $G\backslash Z(G)$, the set of non-central elements of $G$, and
two different vertices $x$ and $y$ are joined by an edge if and
only if $x$ and $y$ do not commute as elements of $G$, i.e.,
$[x,y]\neq 1$. We prove that if $L\in \{L_4(7), L_4(11), L_4(13),
L_4(16), L_4(17)\}$ and $G$ is a finite group such that
$\nabla(G)\cong \nabla(L)$, then $G \cong L$.
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