# Self-centralizing and minimal normal implies monolith

From Groupprops

## 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

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

## Proof

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