October 31st
Today I learned the definition of Pr\"ufer groups. I think the most natural construction is as\[\ZZ\left[\frac1p\right]/\ZZ=\left\{\left\{\frac a{p^n}\right\}:a,n\in\ZZ\right\}=\bigcup_{n=0}^\infty\left\{\sum_{k=0}^n\frac{a_k}{p^k}:a_k\subseteq[0,p)\cap\ZZ\right\}.\]This definition lets us also say that this is $\QQ_p/\ZZ_p$ immediately, which is perhaps a more natural definition. Take $\exp(2\pi i\bullet)$ of this thing, we can say that this group is also isomorphic to\[\left\{\zeta_{p^n}^k:k,n\in\ZZ\right\}.\]With respect to category theory, we note that we have the embeddings $\ZZ/p^n\ZZ\to\ZZ/p^{n+1}\ZZ,$ so we can define $\QQ_p/\ZZ_p$ as\[\varinjlim\ZZ/p^\bullet\ZZ.\]This is seen most obviously by viewing this as $\ZZ[1/p]/\ZZ.$
Wikipedia defines these things as the unique $p$-group for which each element has exactly $p$ $p$th roots. This is indeed true of the group because\[\frac a{p^n}=p\cdot\frac{a+kp^n}{p^{n+1}}\]for any $k,$ of which there are $p$ options. I don't know why this would uniquely define a group; it's not even obvious to me why this property implies that the group is abelian.