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 $g^{m}=h^{n}$ .

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).

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}