Minimal normal implies powering-invariant in solvable group

Statement
Suppose $$G$$ is a solvable group and $$H$$ is a minimal normal subgroup of $$G$$. Then, $$H$$ is a powering-invariant subgroup of $$G$$. Explicitly, for any prime number $$p$$ such that $$G$$ is $$p$$-powered, $$H$$ is also $$p$$-powered.

Facts used

 * 1) uses::Minimal normal implies additive group of a field in solvable group
 * 2) The additive group of a field must be either finite or a rational vector space.
 * 3) uses::Finite implies powering-invariant
 * 4) Any rational vector space is rationally powered, hence powering-invariant in any group containing it.

Proof
The proof follows directy by combining Facts (1)-(4).