Prüfer group

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 that 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:

${\displaystyle \mathbf {Z} (p^{\infty })=\{\exp(2\pi im/p^{n})\mid 0\leq m<p^{n},\,n\in \mathbf {Z} ^{+}\}=\{z\in \mathbf {C} \mid z^{(p^{n})}=1{\text{ for some }}n\in \mathbf {Z} ^{+}\}.\;}$

The group operation here is the multiplication of complex numbers.

There is a presentation

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

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

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:

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

(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).

For each natural number n, consider the quotient group Z/pⁿZ and the embedding Z/pⁿZ → Z/pⁿ⁺¹Z induced by multiplication by p. The direct limit of this system is Z(p^∞):

${\displaystyle \mathbf {Z} (p^{\infty })=\varinjlim \mathbf {Z} /p^{n}\mathbf {Z} .}$

If we perform the direct limit in the category of topological groups, then we need to impose a topology on each of the ${\displaystyle \mathbf {Z} /p^{n}\mathbf {Z} }$, and take the final topology on ${\displaystyle \mathbf {Z} (p^{\infty })}$. If we wish for ${\displaystyle \mathbf {Z} (p^{\infty })}$ to be Hausdorff, we must impose the discrete topology on each of the ${\displaystyle \mathbf {Z} /p^{n}\mathbf {Z} }$, resulting in ${\displaystyle \mathbf {Z} (p^{\infty })}$ to have the discrete topology.

We can also write

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

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

Properties

The complete list of subgroups of the Prüfer p-group Z(p^∞) = Z[1/p]/Z is:

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

Here, each ${\displaystyle \left({1 \over p^{n}}\mathbf {Z} \right)/\mathbf {Z} }$ 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 that 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]

See also

• p-adic integers, which can be defined as the inverse limit of the finite subgroups of the Prüfer p-group.
• Dyadic rational, rational numbers of the form a/2^b. The Prüfer 2-group can be viewed as the dyadic rationals modulo 1.
• Cyclic group (finite analogue)
• Circle group (uncountably infinite analogue)

Notes

References

• Jacobson, Nathan (2009). Basic algebra. Vol. 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) [1994], "Quasi-cyclic group", Encyclopedia of Mathematics, EMS Press

This page was last edited on 7 December 2023, at 01:57