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:

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.

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:

Journal references

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