Divisibility is inherited by extensions where the normal subgroup is contained in the hypercenter

Statement
Suppose we have the following:


 * A group $$G$$.
 * A normal subgroup $$H$$ of $$G$$.
 * A set $$\pi$$ of prime numbers.

such that the following are satisfied:


 * $$H$$ is a normal subgroup contained in the hypercenter in $$G$$ (note that this condition is automatically satisfied if $$G$$ is a nilpotent group).
 * Both $$H$$ and $$G/H$$ are $$\pi$$-divisible groups.

Then, $$G$$ is also a $$\pi$$-divisible group.

Facts used

 * 1) uses::Divisibility is inherited by central extensions
 * 2) Something to show that each of the groups $$[H,G]$$, $$H,G],G]$, and more importantly, each of the quotients between successive groups of this sort, are $\pi$-divisible (we induct from the last one up, in a manner similar to the [[equivalence of definitions of nilpotent group that is divisible for a set of primes.