Characteristically simple and NSCFN implies monolith

Statement

A NSCFN-subgroup of a group that is characteristically simple as a group, must be a monolith: it is a minimal normal subgroup and is contained in every nontrivial normal subgroup.

Facts used

1. Characteristically simple and normal fully normalized implies minimal normal
2. Self-centralizing and minimal normal implies monolith

Proof

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