Let p > 0 is prime number . Let $G$ = $\mathbb Z_{p}$ $\oplus$ $\mathbb Z_{p^2}$ $\oplus$ $\mathbb Z_{p^4}$. Find number of decompositions of G in direct sum of cyclic groups. $A$ $\oplus$ $B$ $\oplus$ $C$, $A \cong \mathbb Z_{p}, B \cong \mathbb Z_{p^2}, C \cong \mathbb Z_{p^4}$
I tried something like this:
- Find number of elements of order $p^4$: $(p^4-p^3)*p^3$. So there are $(p^4-p^3)*p^3/(p^4-p^3) = p^3$ groups $C, C\cong \mathbb Z_{p^4}$.
- Now we should Find $B$. Elements of order $p^2$ is in $\mathbb Z_{p^2}$ and $\mathbb Z_{p^4}$. The direct sum $\mathbb Z_{p^2} \oplus \mathbb Z_{p^4}$ should have zero intersection. I do not how to calculate number of elements of order $p^2$ in this direct sum. Could you please help?
- Find number of elements of order $p$: $(p^3 - 1)-(p^2-1)=p^3-p^2$. Because there is $p^2-1$ elemtns of order $p$ in $\mathbb Z_{p^2} \oplus \mathbb Z_{p^4}$. So there are $(p^3-p^2)/(p-1) = p^2$ groups $A, A \cong \mathbb Z_{p}$.
Final stage is to multiply numbers from 3 items.
Could you please explain how to find number from second item? Any help is highly appreciated.