### Prūfer Group

This is a short note on the Prūfer group.

Let $p$ be a prime integer. The Prūfer group, written as $\Bbb{Z}(p^\infty)$, is the unique $p$-group in which each element has $p$ different $p$th roots. What does this mean? Take $\Bbb{Z}/5\Bbb{Z}$ for example. Can we say that for any element $a$ in this group, there are $5$ mutually different elements which, when raised to the $5$th power, give $a$? No. Take $\overline{2}\in\Bbb{Z}/5\Bbb{Z}$ for instance. We know that only $2$, when raised to the $5$th power, would give $2$. What about $\Bbb{Z}/2^2\Bbb{Z}$? Here $p=2$. Does every element have two mutually different $2$th roots? No. For instance, $\overline{2}\in\Bbb{Z}/2^2\Bbb{Z}$ doesn’t. We start to get the feeling that this condition would only be satisfied in a very special kind of group.

The Prūfer $p$-group may be identified with the subgroup of the circle group $U(1)$, consisting of all the $p^n$-th roots of unity, as $n$ ranges over all non-negative integers. The circle group is the multiplicative group of all complex numbers with absolute value $1$. It is easy to see why this set would be a group. And using the imagery from the circle, it easy to see why each element would have $p$ different $p$th roots. Say we take an element $a$ of the Prūfer group. Assume that it is a $p^{n}$th root of $1$. Then its $p$ different $p$th roots are $p^{n+1}$th roots of $1$. It is nice to see a geometric realization of this rather strange group that seems to rise naturally from groups of the form $\Bbb{Z}/p^n\Bbb{Z}$.