# Normality-large and minimal normal implies monolith

From Groupprops

## Statement

A Minimal normal subgroup (?) of a group that is normality-large, must be a Monolith (?): it is contained in every nontrivial normal subgroup.

## Facts used

## Proof

**Given**: A group , a minimal normal subgroup such that if is trivial for any normal subgroup , then is trivial.

**To prove**: For any normal subgroup , either is trivial or .

**Proof**: Since both and are normal, fact (1) tells us that is normal. So is a normal subgroup of contained in a *minimal* normal subgroup . So there are two cases:

- : In this case, .
- is trivial: In this case, the normality-largeness of tells us that is trivial.

This completes the proof.