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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