Prüfer group

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" "p"th roots.

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

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

*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):$mathbf\{Z\}(p^infty)\; =\; langle\; x\_1\; ,\; x\_2\; ,\; ...\; |\; p\; x\_1\; =\; 0,\; p\; x\_2\; =\; x\_1\; ,\; p\; x\_3\; =\; x\_2\; ,\; ...\; angle$.

*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 [D. L. Armacost and W. L. Armacost, " [http://projecteuclid.org/Dienst/UI/1.0/Summarize/euclid.pjm/1102968274 On "p"-thetic groups] ", "Pacific J. Math.", 41, no. 2 (1972), 295–301] .

*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. [Subgroups of an abelian group are abelian, and coincide with submodules as a $mathbf\{Z\}$-module.] It can thus be used as a counterexample against the idea that every Artinian module is Noetherian (whereas every Artinian "ring" is Noetherian).

ee also

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

References

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