# Minimal normal implies powering-invariant in solvable group

From Groupprops

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.

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## Statement

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

## Facts used

- Minimal normal implies additive group of a field in solvable group
- The additive group of a field must be either finite or a rational vector space.
- Finite implies powering-invariant
- 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).