Prüfer group

The Prüfer 2-group with presentation <g_n: g_n+1² = g_n, g₁² = e>, illustrated as a subgroup of the unit circle in the complex plane.

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

The Prüfer p-groups are countable abelian groups which are important in the classification of infinite abelian groups: they (along with the group of rational numbers) form the smallest building blocks of all divisible groups.

The groups are named after Heinz Prüfer, a German mathematician of the early 20th century.

Constructions of Z(p^∞)

The Prüfer p-group may be identified with the subgroup of the circle group, U(1), consisting of all pⁿ-th roots of unity as n ranges over all non-negative integers:

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

The group operation here is the multiplication of complex numbers.

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

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

(where Z[1/p] denotes the group of all rational numbers whose denominator is a power of p, using addition of rational numbers as group operation).

We can also write

{\mathbf {Z}}(p^{\infty })={\mathbf {Q}}_{p}/{\mathbf {Z}}_{p}

where Q_p denotes the additive group of p-adic numbers and Z_p is the subgroup of p-adic integers.

There is a presentation

{\mathbf {Z}}(p^{\infty })=\langle \,g_{1},g_{2},g_{3},\ldots \mid g_{1}^{p}=1,g_{2}^{p}=g_{1},g_{3}^{p}=g_{2},\dots \,\rangle .

Here, the group operation in Z(p^∞) is written as multiplication.

Properties

The complete list of subgroups of the Prüfer p-group Z(p^∞) is:

0\subset \left({1 \over p}{\mathbf {Z}}\right)/{\mathbf {Z}}\subset \left({1 \over p^{2}}{\mathbf {Z}}\right)/{\mathbf {Z}}\subset \left({1 \over p^{3}}{\mathbf {Z}}\right)/{\mathbf {Z}}\subset \cdots \subset {\mathbf {Z}}(p^{\infty })

(Here $\left({1 \over p^{n}}{\mathbf {Z}}\right)/{\mathbf {Z}}$ is a cyclic subgroup of Z(p^∞) with pⁿ elements; it contains precisely those elements of Z(p^∞) whose order divides pⁿ and corresponds to the set of pⁿ-th roots of unity.) The Prüfer p-groups are the only infinite groups whose subgroups are totally ordered by inclusion. This sequence of inclusions expresses the Prüfer p-group as the direct limit of its finite subgroups. As there is no maximal subgroup of a Prüfer p-group, it is its own Frattini subgroup.

Given this list of subgroups, it is clear that the Prüfer p-groups are indecomposable (cannot be written as a direct sum of proper subgroups). More is true: the Prüfer p-groups are subdirectly irreducible. An abelian group is subdirectly irreducible if and only if it is isomorphic to a finite cyclic p-group or to a Prüfer group.

The Prüfer p-group is the unique infinite p-group which is locally cyclic (every finite set of elements generates a cyclic group). As seen above, all proper subgroups of Z(p^∞) are finite. The Prüfer p-groups are the only infinite abelian groups with this property.^[1]

The Prüfer p-groups are divisible. They play an important role in the classification of divisible groups; along with the rational numbers they are the simplest divisible groups. More precisely: an abelian group is divisible if and only if it is the direct sum of a (possibly infinite) number of copies of Q and (possibly infinite) numbers of copies of Z(p^∞) for every prime p. The (cardinal) numbers of copies of Q and Z(p^∞) that are used in this direct sum determine the divisible group up to isomorphism.^[2]

As an abelian group (that is, as a Z-module), Z(p^∞) is Artinian but not Noetherian.^[3] It can thus be used as a counterexample against the idea that every Artinian module is Noetherian (whereas every Artinian ring is Noetherian).

The endomorphism ring of Z(p^∞) is isomorphic to the ring of p-adic integers Z_p.^[4]

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.^[5]

Notes

↑ See Vil'yams (2001)
↑ See Kaplansky (1965)
↑ See also Jacobson (2009), p. 102, ex. 2.
↑ See Vil'yams (2001)
↑ D. L. Armacost and W. L. Armacost,"On p-thetic groups", Pacific J. Math., 41, no. 2 (1972), 295–301

References

Jacobson, Nathan (2009). Basic algebra. 2 (2nd ed.). Dover. ISBN 978-0-486-47187-7
Pierre Antoine Grillet (2007). Abstract algebra. Springer. ISBN 978-0-387-71567-4.
Kaplansky, Irving (1965). Infinite Abelian Groups. University of Michigan Press.
N.N. Vil'yams (2001), "Quasi-cyclic group", in Hazewinkel, Michiel, Encyclopedia of Mathematics, Springer, ISBN 978-1-55608-010-4

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Prüfer group

Constructions of Z(p∞)

Properties

See also

Notes

References

Constructions of Z(p^∞)