Prüfer group

From Wikipedia, the free encyclopedia

The Prüfer 2-group. <gn: gn+12 = gn, g12 = e>

The Prüfer 2-group. <g_n: g_n+1² = g_n, g₁² = e>

In mathematics, specifically group theory, the Prüfer p-group or the p-quasicyclic group or p^∞-group, Z(p^∞), for a prime number p is the unique torsion group in which every element has p pth roots.

The Prüfer p-group may be represented as a subgroup of the circle group, U(1), as the set of pⁿth roots of unity as n ranges over all non-negative integers:

$\mathbf{Z}(p^\infty)=\{\exp(2\pi i n/p^m) \mid n\in \mathbf{Z}^+,\,m\in \mathbf{Z}^+\}\;$

Alternatively, the Prüfer p-group may be seen as the Sylow p-subgroup of Q/Z, consisting of those elements whose order is a power of p:

$\mathbf{Z}(p^\infty) = \mathbf{Z}[1/p]/\mathbf{Z}$

There is a presentation (written additively)

$\mathbf{Z}(p^\infty) = \langle x_1 , x_2 , ... | p x_1 = 0, p x_2 = x_1 , p x_3 = x_2 , ...\rangle$ .

The Prüfer p-group is the unique infinite p-group which is locally cyclic (every finite set of elements generates a cyclic group).

The Prüfer p-group is divisible.

In the theory of locally compact topological groups the Prüfer p-group (endowed with the discrete topology) is the Pontryagin dual of the compact group of p-adic integers, and the group of p-adic integers is the Pontryagin dual of the Prüfer p-group^[1].

The Prüfer p-groups for all primes p are the only infinite groups whose subgroups are totally ordered by inclusion. As there is no maximal subgroup of a Prüfer p-group, it is its own Frattini subgroup.

$0 \subset \mathbf{Z}/p \subset \mathbf{Z}/p^2 \subset \mathbf{Z}/p^3 \subset \cdots \subset \mathbf{Z}(p^\infty)$

This sequence of inclusions expresses the Prüfer p-group as the direct limit of its finite subgroups.

As a $\mathbf{Z}$ -module, the Prüfer p-group is Artinian, but not Noetherian, and likewise as a group, it is Artinian but not Noetherian.^[2] It can thus be used as a counterexample against the idea that every Artinian module is Noetherian (whereas every Artinian ring is Noetherian).

[edit] See also

p-adic integers, which can be defined as the inverse limit of the finite subgroups of the Prüfer p-group.

[edit] References

^ D. L. Armacost and W. L. Armacost, "On p-thetic groups", Pacific J. Math., 41, no. 2 (1972), 295–301
^ Subgroups of an abelian group are abelian, and coincide with submodules as a $\mathbf{Z}$ -module.