Minimal normal implies powering-invariant in solvable group

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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 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).