Normality-large and minimal normal implies monolith

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

1. Normality is strongly intersection-closed

Proof

Given: A group $G$, a minimal normal subgroup $N$ such that if $N\cap M$ is trivial for any normal subgroup $M$, then $M$ is trivial.

To prove: For any normal subgroup $M$, either $M$ is trivial or $N\leq M$.

Proof: Since both $N$ and $M$ are normal, fact (1) tells us that $N\cap M$ is normal. So $N\cap M$ is a normal subgroup of $G$ contained in a minimal normal subgroup $N$. So there are two cases:

• $N\cap M=N$: In this case, $N\leq M$.
• $N\cap M$ is trivial: In this case, the normality-largeness of $N$ tells us that $M$ is trivial.

This completes the proof.