  A signed dominating function of graph $\Gamma$ is a function $g :V(\Gamma)
\longrightarrow \{-1,1\}$ such that $\sum_{u \in N[v]}g(u) >0$ for each $v \in
V(\Gamma)$. The signed domination number $\gamma_{_S}(\Gamma)$ is the minimum
weight of a signed dominating function on $\Gamma$. Let $G=\langle S \rangle$
be a finite group such that $e \not\in S=S^{-1}$. In this paper, we obtain the
signed domination number of $Cay(S:G)$ based on cardinality of $S$. Also we
determine the classification of group $G$ by $|S|$ and $\gamma_{_S}(Cay(S:G))$.
