Divisible group for a set of primes

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Definition

Let \pi be a set of primes. A group G is termed \pi-divisible if it satisfies the following equivalent definitions:

No. Shorthand Explanation
1 p-divisible, for each prime p \in \pi For every g \in G and every p \in \pi, there exists h \in G such that h^p = g. In other words, the map x \mapsto x^p is surjective from G to itself for all p \in \pi.
2 n-divisible for every \pi-number n if g \in G and n is a natural number all of whose prime divisors are in the set \pi, then there exists an element h \in G such that h^n = g.
3 (not necessarily unique) rational powers with denominators \pi-number if g \in G and m,n are integers with all prime divisors of n in \pi, there exists h \in G satisfying g^m = h^n.

Related notions

Conjunction with group properties

References

Journal references

  • Some aspects of groups with unique roots by Gilbert Baumslag, Acta mathematica, Volume 104, Page 217 - 303(Year 1960): PDF (ungated)More info: This paper uses the notation E_{\pi}-group to describe what we call a \pi-divisible group. The notation is introduced in Section 2, Page 218 (second page of the paper).

Textbook references

  • p-automorphisms of finite p-groups by Evgenii I. Khukhro, 13-digit ISBN 978-0-521-59717-3, 10-digit ISBN 0-521-59717-X, Page 18, Section 1.3 (Algebraic systems, varieties, and free objects), More info