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Let $S$ be the set of all $3$-cycles in $S_n$. The \\emph{complete alternating group graph}, denoted by $CAG_n$, is defined as the Cayley graph $\\mathrm{Cay}(A_n,S)$ on $A_n$ with respect to $S$. In this paper, we show that $CAG_n$ ($n\\geq 4$) is not a normal Cayley graph. 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