Minimal normal implies powering-invariant in solvable group

This article gives the statement and possibly, proof, of an implication relation between two subgroup properties, when the big group is a solvable group. That is, it states that in a Solvable group (?), every subgroup satisfying the first subgroup property (i.e., Minimal normal subgroup (?)) must also satisfy the second subgroup property (i.e., Powering-invariant subgroup (?)). In other words, every minimal normal subgroup of solvable group is a powering-invariant subgroup of solvable group.
View all subgroup property implications in solvable groups $|$ View all subgroup property non-implications in solvable groups $|$ View all subgroup property implications $|$ View all subgroup property non-implications

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. 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. 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).