Self-centralizing and minimal normal implies monolith

Statement

Any Minimal normal subgroup (?) of a group that is self-centralizing, is a Monolith (?): it is the unique minimal normal subgroup, and is contained in every minimal normal subgroup.

Related facts

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. Self-centralizing and normal implies normality-large
2. Normality-large and minimal normal implies monolith

Proof

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