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Let $\$_n$ be the symmetric group on $n$ letters and $G$ be a subgroup of
$\$_n$. Suppose $\chi$ is an irreducible complex character of $G$ and $d_{\chi}^G$ is the generalized matrix function afforded by $G$ and $\chi$. In this paper we characterize for which $G$ and $\chi$, the Cayley-Hamilton Theorem holds.
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