1. Let the function $f: \mathbb{Z}_{9} \longrightarrow \mathbb{Z}_{9}, f(x)=x^{9}$. Determine the non-empty subsets $A$ of the set $\mathbb{Z}_{9}$ with the property $f(A)=A$.
Solution. The invertible elements of the ring $Z_{9}$ form the set $U=\{\hat{1}, \hat{2}, \hat{4}, \hat{5}, \hat{7}, \hat{8}\}$.

$(U, \cdot)$ is an abelian group, so $x^{6}=\hat{1}, \forall x \in U$. Thus, $f(x)=x^{3}, \forall x \in U$. We obtain $f(\hat{1})=f(\hat{4})=$ $f(\hat{7})=\hat{1}, f(\hat{2})=f(\hat{5})=f(\hat{8})=\hat{8}$ and $f(\hat{0})=f(\hat{3})=f(\hat{6})=\hat{0} .(3 \mathbf{p})$

We verify the property $f(A)=A \Leftrightarrow A \subset f\left(\mathbb{Z}_{9}\right)=\{\hat{0}, \hat{1}, \hat{8}\}$, so the non-empty subsets required are: $\{\hat{0}\},\{\hat{1}\},\{\hat{8}\},\{\hat{0}, \hat{1}\},\{\hat{0}, \hat{8}\},\{\hat{1}, \hat{8}\},\{\hat{0}, \hat{1}, \hat{8}\} .(\mathbf{p} \mathbf{p})$