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The p-adic number system for any prime number p extends the ordinary arithmetic of the rational numbers in a different way from the extension of the rational number system to the real and complex number systems

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Can an abelian group have multiple different actions of $\mathbb{Z}_p$?

This is perhaps a trivial question, but I've asked a few colleagues and they couldn't answer. For a given abelian group $M$, is it possible to have several different actions of the ring of $p$-adic nu …
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