(adsbygoogle = window.adsbygoogle || []).push({}); 1. The problem statement, all variables and given/known data

Show that [itex]\mathbb{Z}_{p^r}[p][/itex] is isomorphic to [itex]\mathbb{Z}_p[/itex] for any [itex]r \geq 1[/itex] and prime p.

[itex]\mathbb{Z}_{p^r}[p][/itex] is defined as the subgroup [itex]\{x \in \mathbb{Z}_{p^r} | px = 0 \}[/itex]

2. Relevant equations

3. The attempt at a solution

I don't think I should need to use Sylow's Theorems for this since it is in a different section. I can only think of two elements in that subgroup p^{r-1} and 0 and am not really sure how to find the rest or figure out their subalgebra. Actually I guess I just need to prove that the group has p elements and then the only possibility will be Z_p. But how to do that?

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# Free abelian groups

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