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 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.

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:

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:

(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

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

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

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

(Here ${\displaystyle \left({1 \over p^{n}}\mathbf {Z} \right)/\mathbf {Z} }$ 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.

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