Characteristically simple and NSCFN implies monolith

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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).