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We show that if a finite group $G$ has trivial center and $N(G)$ equals to $N(Alt_n)$ or $N(Sym_n)$ for $n\\geq 23$, then $G$ has a composition factor isomorphic to an alternating group $Alt_k$ such that $k\\leq n$ and the half-interval $(k, n]$ contains no primes. 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