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May I ask someone elaborate how to treat the quotient group $\frac{p\mathbb Z}{p^\alpha\mathbb Z}$ when $p$ is prime? Any answer is appreciated.

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Consider the composition of homomorphisms of additive groups $$ \Bbb Z\longrightarrow p\Bbb Z\longrightarrow p\Bbb Z/p^a\Bbb Z $$ where the first one is defined by declaring that the image of $1$ is $p$ and the second is the quotient map.

The composite homomorphism is clearly surjective and its kernel is $p^{a-1}\Bbb Z$ (why?). Thus we get an isomorphism $$ \frac{\Bbb Z}{p^{a-1}\Bbb Z}\simeq\frac{p\Bbb Z}{p^a\Bbb Z} $$ by the first isomorphism theorem.

Mind that the fact that $p$ is prime has no role in the above.

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