Find number of decompositions of $G = \mathbb{Z}_p\oplus\mathbb Z_{p^2}\oplus\mathbb Z_{p^3}$ in cyclic groups of given orders. (Here $p$ is prime.)
As I understood I need to find such $A\oplus B\oplus C, $ that $ A\cong \mathbb Z_p, B\cong \mathbb Z_{p^2}, C\cong \mathbb Z_{p^3}$. So I need to find all of the generators of these cyclic groups?