Normality-large and minimal normal implies monolith

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

This completes the proof.