Prüfer group

Example

Let $p$ be a prime. The Prüfer $p$-group is $$\Z(p^\infty) = \Z[1/p] / \Z = \set{m/p^k + \Z : m \in \Z, \ k \ge 0},$$ a subgroup of $\Q/\Z$. Equivalently, it is the group of all complex roots of unity of order a power of $p$, a subgroup of the circle group.

It is a divisible group. Let $x = m/p^k + \Z$ and $n \ge 1$, and write $n = p^j r$ with $p \nmid r$. Choose an integer $s$ with $rs \equiv 1 \pmod{p^{k+j}}$, and put $y = sm/p^{k+j} + \Z$. Then $ny = rs \, m / p^k + \Z$, and $rs \, m/p^k - m/p^k = (rs - 1) m / p^k$ is an integer, because $p^{k+j}$ divides $rs - 1$; so $ny = x$.