Prüfer group

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In the theory of abelian groups the Prüfer p-group or the p-quasicyclic group or p∞-group, Z(p∞), for a prime number p is the unique torsion group in which every element has p pth roots.

  • The Prüfer p-group may be represented as a subgroup of the circle group, U(1), as the set of pnth roots of unity as n ranges over all non-negative integers:
\mathbb{Z}(p^\infty)=\{e^\frac{2 n i\pi}{p^m}\,|\,n\in \mathbb{Z}^+,\,m\in \mathbb{Z}^+\}\;
  • The Prüfer p-group is the unique infinite p-group which is locally cyclic (every finite set of elements generates a cyclic group).
  • The Prüfer p-groups for all primes p are the only infinite groups whose subgroups are totally ordered by inclusion. As there is no maximal subgroup of a Prüfer p-group, it is its own Frattini subgroup.
\mathbb{Z}_1 \subseteq \mathbb{Z}_p \subseteq \mathbb{Z}_{p^2} \subseteq \mathbb{Z}_{p^3} \subseteq \ldots\ \subset \mathbb{Z}(p^\infty)
This sequence of inclusions expresses the Prüfer p-group as the direct limit of its finite subgroups.

[edit] References

1 D. L. Armacost and W. L. Armacost, On p-thetic groups, Pacific J. Math. 41, no. 2 (1972), 295–301

[edit] External links

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