Powered group for a set of primes

Definition

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

No.	Shorthand	Explanation
1	$p$-powered, or uniquely $p$-divisible, for each prime $p\in \pi$	For every $g\in G$ and every $p\in \pi$, there is a unique $h\in G$ such that $h^{p}=g$.
2	unique rational powers with denominators $\pi$-numbers	Given any integers $m,n$, with $n$ a $\pi$-number (viz., a nonzero integer all of whose prime factors are in $\pi$), and any $g\in G$, there exists a unique $h\in G$ such that $g^{m}=h^{n}$.
3	group powered over the ring $\mathbb {Z} [\pi ^{-1}]$	$G$ is a group powered over the ring $\mathbb {Z} [\pi ^{-1}]$, i.e., the ring obtained by adjoining inverses of all elements of $\pi$ to $G$.

A rationally powered group is a group powered for the set of all primes.

The special case of nilpotent groups

In the case of nilpotent groups, we have the following result: nilpotent group is powered over a prime iff its abelianization is.