On the number of symmetric coordinate functions of APN function

In the paper, symmetric properties of APN functions are considered. For any APN function $F$, it is proved that $F$ can not be a symmetric vector function and there is no permutation of its coordinates such that $F$ keeps its value. Theorems about strict upper bounds for the number of its symmetric and rotation symmetric coordinate Boolean functions are proved. The lower bound for the number of distinct values of $F$ is obtained. It is shown that there exists an upper bound for the maximal number of coinciding values of $F$.