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For questions about groups whose elements commute.

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How many elements of each order are there in this $p$-group?

We have $m=\phi(p^{k+1})=p^k(p-1)$. Since the orders of all cyclic factors $<p^k$, the number of Prufer factors should be at least 1, at least one coordinate of every element of order $p^{k+1}$ must …
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