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Let $$G={ {<a>}_{p} \times {<b>}_{p} \times {<c>}_{p^2}} \cong \mathbb{Z}_p \times \mathbb{Z}_p \times \mathbb{Z}_{p^2} \text{, $p$ is prime}$$ There are $p^3-1$ elements with order $p$, $p^2(p^2-p)$ elements with order $p^2$.

So my questions are:

1) How many cyclic subgroups order $p^2$ (like $\mathbb{Z}_{p^2}, \text{ not } \mathbb{Z}_p \times \mathbb{Z}_p$) are in G? (As i know, number of all subgroups order $p^2$ in G are $2p^2+p+1$)

2) How many subgroups order $p$ are in $\mathbb{Z}_p \times \mathbb{Z}_{p^2}$

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Lemma: Let $n$ be number of the elements of order $m$ then there are $$\dfrac{n}{\phi(m)}$$ cylic subgroup of order $m$.

By the lemma, There are $\dfrac{p^2(p^2-p)}{\phi(p^2)}=p^2$ cyclic subgroup of order $p^2$.

I left the second question to you.

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