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Let A be finite commutative group say $(Z_m)^h$. I will say that $S \subset A$ is an orbit if exist group $H$ which acts on A such that $S$ is an orbit of $H$.

Can one give a simple characterization of all orbits of $(Z_m)^h$?

By action on $A$ I mean automorphisms of a group A.

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Let A be finite commutative group say $(Z_m)^h$. I will say that $S \subset A$ is an orbit if exist group $H$ which acts on A such that $S$ is an orbit of $H$.

Can one give a simple characterization of all orbits of $(Z_m)^h$?

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