3. Determine the subsets $A$ of $\mathbb{Z}_{6}$ with the property that $A$ has three elements and there exists a function $f: A \rightarrow A, f(x)=x+a, a \in \mathbb{Z}_{6} \backslash\{\hat{0}\}$.

Prof. Bene Marius, Alexandria
3. $A=\left\{x_{1}, x_{2}, x_{3}\right\}, f: A \rightarrow A, f(x)=x+a$ is injective

$A$ being finite implies $f$ is bijective and $f\left(x_{1}\right)+f\left(x_{2}\right)+f\left(x_{3}\right)=x_{1}+x_{2}+x_{3}$

$x_{1}+x_{2}+x_{3}+\hat{3} a=x_{1}+x_{2}+x_{3}$, thus $\widehat{3} a=\hat{0}, a \in \mathbb{Z}_{6} \backslash\{\hat{0}\}$,

it follows that $a \in\{\hat{2}, \hat{4}\}$. For $a=\hat{2}, f(x)=x+\hat{2}$, it results in $A=\{\hat{0}, \hat{2}, \hat{4}\}$ and

$A=\{\hat{1}, \hat{3}, \hat{5}\}$, and for $a=\hat{4}, f(x)=x+\hat{4}$, it results in $A=\{\hat{0}, \hat{2}, \hat{4}\}$,

$A=\{\hat{1}, \hat{3}, \hat{5}\}$