# Characteristically simple and NSCFN implies monolith

From Groupprops

## 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

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

## Proof

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