for any integer $n\ge2$, consider the group ${\bf Z}_{p^n}\oplus{\bf Z}_p$. Determine the number of cyclic subgroups of order $p$

Question

I know ${\bf Z}_p\oplus{\bf Z}_p$ has a cyclic group of p+1 order p. but what about when it's n? (p is prime)

Answer 1

The elements of ${\bf Z}_{p^n}\oplus{\bf Z}_p$ are the ordered pairs $(a,b)$ with $a$ in ${\bf Z}_{p^n}$ and $b$ in ${\bf Z}_p$. Assuming here and throughout that $n\ge1$, the number of elements of order 1 or $p$ in ${\bf Z}_{p^n}$ is $p$, so the number of elements of order 1 or $p$ in ${\bf Z}_{p^n}\oplus{\bf Z}_p$ is $p^2$. Exactly one of these elements has order 1, so there are $p^2-1$ elements of order $p$ in ${\bf Z}_{p^n}\oplus{\bf Z}_p$.

Each subgroup of order $p$ contains $p-1$ of these elements of order $p$, and each element of order $p$ is in exactly one subgroup of order $p$, so the number of subgroups of order $p$ is $(p^2-1)/(p-1)$, which is $p+1$.