Prüfer group

Short description: Mathematical term in group theory

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 $p^{\mathrm{\infty }}$-group, $\mathbb{\mathbb{Z}}(p^{\mathrm{\infty }})$, 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

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

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

The group operation here is the multiplication of complex numbers.

There is a presentation

$\mathbb{\mathbb{Z}}(p^{\mathrm{\infty }})=⟨g_1,g_2,g_3,\dots \mid g_1^p=1,g_2^p=g_1,g_3^p=g_2,\dots ⟩.$

Here, the group operation in $\mathbb{\mathbb{Z}}(p^{\mathrm{\infty }})$ is written as multiplication.

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

$\mathbb{\mathbb{Z}}(p^{\mathrm{\infty }})=\mathbb{\mathbb{Z}}[1/p]/\mathbb{\mathbb{Z}}$

(where $\mathbb{\mathbb{Z}}[1/p]$ 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 $\mathbb{\mathbb{Z}}/p^n\mathbb{\mathbb{Z}}$ and the embedding $\mathbb{\mathbb{Z}}/p^n\mathbb{\mathbb{Z}}\to \mathbb{\mathbb{Z}}/p^{n+1}\mathbb{\mathbb{Z}}$ induced by multiplication by p. The direct limit of this system is $\mathbb{\mathbb{Z}}(p^{\mathrm{\infty }})$:

$\mathbb{\mathbb{Z}}(p^{\mathrm{\infty }})=\underrightarrow{\lim }\mathbb{\mathbb{Z}}/p^n\mathbb{\mathbb{Z}}.$

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

We can also write

$\mathbb{\mathbb{Z}}(p^{\mathrm{\infty }})={\mathbb{\mathbb{Q}}}_p/{\mathbb{\mathbb{Z}}}_p$

where ${\mathbb{\mathbb{Q}}}_p$ denotes the additive group of p-adic numbers and ${\mathbb{\mathbb{Z}}}_p$ is the subgroup of p-adic integers.

Properties

The complete list of subgroups of the Prüfer p-group $\mathbb{\mathbb{Z}}(p^{\mathrm{\infty }})$ is:

$0⊊\left(\frac{1}{p}\mathbb{\mathbb{Z}}\right)/\mathbb{\mathbb{Z}}⊊\left(\frac{1}{p^2}\mathbb{\mathbb{Z}}\right)/\mathbb{\mathbb{Z}}⊊\left(\frac{1}{p^3}\mathbb{\mathbb{Z}}\right)/\mathbb{\mathbb{Z}}⊊\cdots ⊊\mathbb{\mathbb{Z}}(p^{\mathrm{\infty }})$

Here, each $\left(\frac{1}{p^n}\mathbb{\mathbb{Z}}\right)/\mathbb{\mathbb{Z}}$ is a cyclic subgroup of $\mathbb{\mathbb{Z}}(p^{\mathrm{\infty }})$ with pⁿ elements; it contains precisely those elements of $\mathbb{\mathbb{Z}}(p^{\mathrm{\infty }})$ 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 $\mathbb{\mathbb{Z}}(p^{\mathrm{\infty }})$ are finite. The Prüfer p-groups are the only infinite abelian groups with this property.

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 $\mathbb{\mathbb{Q}}$ and (possibly infinite) numbers of copies of $\mathbb{\mathbb{Z}}(p^{\mathrm{\infty }})$ for every prime p. The (cardinal) numbers of copies of $\mathbb{\mathbb{Q}}$ and $\mathbb{\mathbb{Z}}(p^{\mathrm{\infty }})$ that are used in this direct sum determine the divisible group up to isomorphism.

As an abelian group (that is, as a Z-module), $\mathbb{\mathbb{Z}}(p^{\mathrm{\infty }})$ is Artinian but not Noetherian. 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 $\mathbb{\mathbb{Z}}(p^{\mathrm{\infty }})$ is isomorphic to the ring of p-adic integers ${\mathbb{\mathbb{Z}}}_p$.

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.

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)

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. 
• Hazewinkel, Michiel, ed. (2001), "Quasi-cyclic group", Encyclopedia of Mathematics, Springer Science+Business Media B.V. / Kluwer Academic Publishers, ISBN 978-1-55608-010-4, https://www.encyclopediaofmath.org/index.php?title=Q/q076440