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 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

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

1. See Vil'yams (2001)
2. See Kaplansky (1965)
3. See also Jacobson (2009), p. 102, ex. 2.
4. See Vil'yams (2001)
5. 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. 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