Divisible group for a set of primes

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

• Powered group for a set of primes
• Powering-injective group for a set of primes
• Torsion-free group for a set of primes

Conjunction with group properties

• Nilpotent group that is divisible for a set of primes is the conjunction with the property of being a nilpotent group.

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}