## Count the number of cyclic subgroups in an elementary abelian p-group

Let $p$ be a prime and let $n$ be a positive integer. Find a formula for the number of subgroups of order $p$ in the elementary abelian group $E_{p^n}$.

By definition, every nonidentity element of $E_{p^n}$ has order $p$. Thus every nonidentity element generates an order $p$ subgroup. On the other hand, every order $p$ subgroup has $p-1$ generators. Thus the number of distinct order $p$ subgroups is $(p^n-1)/(p-1)$, which is evidently equal to $\sum_{k=0}^{n-1} p^k$.

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