Self-centralizing and minimal normal implies characteristic

This article gives the statement and possibly, proof, of an implication relation between two subgroup properties. That is, it states that every subgroup satisfying the first subgroup property (i.e., self-centralizing minimal normal subgroup) must also satisfy the second subgroup property (i.e., characteristic subgroup)
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Statement

Any Minimal normal subgroup (?) of a group that is self-centralizing must be characteristic.

Related facts

Stronger facts

• Self-centralizing and minimal normal implies strictly characteristic
• Self-centralizing and minimal normal implies monolith

Corollaries

• Characteristically simple and Abelian implies characteristic in holomorph
• Characteristically simple and non-Abelian implies characteristic in automorphism group

Facts used

1. Self-centralizing and minimal normal implies monolith: Any slef-centralizing minimal normal subgroup is contained in every nontrivial normal subgroup.
2. Monolith is characteristic

Proof

This follows together by piecing facts (1) and (2).