Powered group for a set of primes

Definition

Let $\pi$ be a set of primes. A group $G$ is termed $\pi$-powered or uniquely $\pi$-divisible 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 $h^p = g$. In other words, the map $x \mapsto x^p$ is a bijection from $G$ to itself.
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 $\mathbb{Z}$, the ring of integers. (see also additive group of ring of integers localized at a set of primes).
4	$n$-powered for any one number $n$ whose set of prime divisors is precisely $\pi$ (note that this definition can be used only in the case that $\pi$ is a finite set)	There exists a natural number $n$ such that the set of prime divisors of $n$ is exactly $\pi$ and the map $x \mapsto x^n$ is bijective from $G$ to itself.

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

Localization definition

Let $\pi$ be a set of primes. We call a group $G$ a $\pi$-local group if $G$ is powered over all primes not in $\pi$. Note that $G$ may also happen to be powered over one or more of the primes in $\pi$.

Note in particular that a $p$-local group is a group that is powered over all primes other than $p$. Note that this definition differs completely from the definition of p-local subgroup.

Powering versus localization terminology

The domains of abstract group theory as well as combinatorial and geometric group theory often use the "powering" or "unique divisibility" jargon. The domain of algebraic topology typically uses the "localization" jargon because of the connection with local nilpotent spaces.

Conjunction with group properties

Conjunction	Other component of conjunction	Facts about the conjunction
nilpotent group that is powered over a set of primes	nilpotent group

Related notions

We can think of forgetful functors:

category of $\pi$-powered groups $\to$ category of groups $\to$ category of sets

Note that each of these categories arises from a variety of algebras, and the forgetful functors are variety reduct maps.

Both the two individual functors and their composite have left-adjoint functors. The three left-adjoint functors we obtain are:

Functor source	Functor target	Left adjoint to ...	Description of functor
category of groups	category of $\pi$-powered groups	forgetful functor from $\pi$-powered groups to groups	‎Every group admits an initial homomorphism to a pi-powered group. The functor sending a group to the image of this initial homomorphism is termed the $\pi$-powering functor, or equivalently, it is termed the localization functor for the complement of $\pi$ in the set of primes.
category of sets	category of $\pi$-powered groups	forgetful functor from $\pi$-powered groups to sets	See free powered group for a set of primes. A detailed existence proof is at there exist free powered groups for any set of primes and any size of generating set.
category of sets	category of groups	forgetful functor from groups to sets	(abstract) free group functor

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}: Baumslag's paper uses the notation $D_{\pi}$-group for a group powered over the prime set $\pi$.

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}