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  "body": "\n## Example\n\nLet $p$ be a [[prime|01826KMG2H]].\nThe *Prüfer $p$-group* is\n$$\\Z(p^\\infty) = \\Z[1/p] / \\Z = \\set{m/p^k + \\Z : m \\in \\Z, \\ k \\ge 0},$$\na subgroup of $\\Q/\\Z$.\nEquivalently, it is the group of all complex [[roots of unity|0S61N75H45]] of order a power of $p$, a subgroup of the [[circle group|04EAFVFTF7]].\n\nIt is a [[divisible group|ZXVBEPMTCN]].\nLet $x = m/p^k + \\Z$ and $n \\ge 1$, and write $n = p^j r$ with $p \\nmid r$.\nChoose an integer $s$ with $rs \\equiv 1 \\pmod{p^{k+j}}$, and put $y = sm/p^{k+j} + \\Z$.\nThen $ny = rs \\, m / p^k + \\Z$, and $rs \\, m/p^k - m/p^k = (rs - 1) m / p^k$ is an integer, because $p^{k+j}$ divides $rs - 1$; so $ny = x$.\n\n## Remarks\n\nThe Prüfer group is the union of the chain of cyclic groups $\\Z/p\\Z \\subset \\Z/p^2\\Z \\subset \\cdots$, where $\\Z/p^k\\Z$ is identified with $\\set{m/p^k + \\Z}$.\nIts proper subgroups are exactly these finite cyclic groups, so it is not finitely generated.\nIt is a [[p-group|06QYRTX9KQ]] in the sense that every element has order a power of $p$.\nTogether with $\\Q$, the Prüfer groups are the building blocks of all divisible groups.\nIts Pontryagin dual is the group of [[p-adic integers|V6TJAFZ4MD]].\n\n## Explanations\n\n### L1 human\n\nTake the circle and mark the points that you reach by rotating by $1/p$ of a turn, by $1/p^2$ of a turn, and so on.\nAdd all their combinations.\nThis set of rotations is the Prüfer group.\nEvery rotation in it can be split into $n$ equal smaller rotations that still lie in it.\nIt is infinite, but each of its proper subgroups is finite.\n",
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