Orbital derivatives on residue rings. Part I. General properties

For mappings $f\colon H\to F$, where $H$ and $F$ are Abelian groups, a definition of the $t^{th}$-order orbital derivative is introduced. The definition is based on structures of orbits of subgroups of $H$. Properties of the $t^{th}$-order orbital derivative on the residue ring $\mathbb Z_{2^n}$ are described.