This is a short note on the Prūfer group.
Let 
be a prime integer. The Prūfer group, written as 
, is the unique -group in which each element has different th roots. What does this mean? Take 
for example. Can we say that for any element 
in this group, there are 
mutually different elements which, when raised to the th power, give ? No. Take 
for instance. We know that only 
, when raised to the th power, would give . What about 
? Here 
. Does every element have two mutually different th roots? No. For instance, 
doesn’t. We start to get the feeling that this condition would only be satisfied in a very special kind of group.
The Prūfer -group may be identified with the subgroup of the circle group 
, consisting of all the 
-th roots of unity, as 
ranges over all non-negative integers. The circle group is the multiplicative group of all complex numbers with absolute value 
. 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 different th roots. Say we take an element of the Prūfer group. Assume that it is a 
th root of . Then its different th roots are 
th roots of . It is nice to see a geometric realization of this rather strange group that seems to rise naturally from groups of the form 
.
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The Prūfer groups are fun objects. They have lots of cool properties. For instance, they are the only infinite groups whose subgroups are totally ordered by inclusion. They also serve as a nice counterexample to the (false) claim that a group with only finite subgroups must be finite.