Self-centralizing and minimal normal implies monolith

Statement
Any fact about::minimal normal subgroup of a group that is self-centralizing, is a fact about::monolith: it is the unique minimal normal subgroup, and is contained in every minimal normal subgroup.

Applications

 * Characteristically simple and NSCFN implies monolith
 * Characteristically simple and Abelian implies monolith in holomorph
 * Characteristically simple and non-Abelian implies monolith in automorphism group

Facts used

 * 1) uses::Self-centralizing and normal implies normality-large
 * 2) uses::Normality-large and minimal normal implies monolith

Proof
The proof follows directly by combining facts (1) and (2).