Characteristic subgroup of uniquely p-divisible abelian group is uniquely p-divisible

For a single prime number
Suppose $$p$$ is a prime number and $$G$$ is a uniquely $$p$$-divisible fact about::abelian group. In other words, for any $$x \in G$$, there is a unique $$y \in G$$ such that $$py = x$$. Suppose $$H$$ is a uses property satisfaction of::characteristic subgroup of $$G$$. Then, $$H$$ is also uniquely $$p$$-divisible.

For a collection of primes
Suppose $$\pi$$ is a set of primes (possibly infinite) and $$G$$ is an abelian group that is powered over $$\pi$$, i.e., it is uniquely $$p$$-divisible for all $$p \in \pi$$. Suppose $$H$$ is a characteristic subgroup of $$G$$. Then, $$H$$ is also powered over $$\pi$$.

In particular, taking the case where $$\pi$$ is the set of all primes, this states that a characteristic subgroup of a uses property satisfaction of::rationally powered group is also a proves property satisfaction of::rationally powered group.

Facts used

 * 1) uses::abelian implies uniquely p-divisible iff pth power map is automorphism