Self-centralizing and minimal normal implies characteristic

Statement
Any fact about::minimal normal subgroup of a group that is self-centralizing must be characteristic.

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

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