Prūfer Group

by ayushkhaitan3437

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}.

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