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 pth 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 5th power, give a? No. Take \overline{2}\in\Bbb{Z}/5\Bbb{Z} for instance. We know that only 2, when raised to the 5th power, would give 2. What about \Bbb{Z}/2^2\Bbb{Z}? Here p=2. Does every element have two mutually different 2th 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 pth 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 pth 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}.

Published by ayushkhaitan3437

Hello! My name is Ayush Khaitan, and I'm a graduate student in Mathematics.

One thought on “Prūfer Group

  1. 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.

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