Normality-large and minimal normal implies monolith
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.