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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