Monolith is characteristic

This article gives the statement and possibly, proof, of an implication relation between two subgroup properties. That is, it states that every subgroup satisfying the first subgroup property (i.e., monolith) must also satisfy the second subgroup property (i.e., strictly characteristic subgroup)
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Statement

Verbal statement

If a group has a monolith (a Minimal normal subgroup (?) contained in every nontrivial normal subgroup), then that monolith is a characteristic subgroup (it is invariant under any automorphism of the group).

Related facts

Stronger facts

• Monolith is strictly characteristic
• Monolith is fully characteristic in finite groups

Applications

• Self-centralizing and minimal normal implies characteristic

Facts used

1. Normality satisfies inverse image condition

Proof

Given: A group $G$, a minimal normal subgroup $N$ such that $N\le M$ for any nontrivial normal subgroup $M$. An automorphism $\sigma$ of $G$.

To prove: $\sigma (N)\le N$.

Proof: Consider the subgroup ${\sigma }^{-1}(N)$. This is normal by fact (1), either ${\sigma }^{-1}(N)$ is trivial or $N\le {\sigma }^{-1}(N)$. Since $\sigma$ is surjective and $N$ is nontrivial, ${\sigma }^{-1}(N)$ cannot be trivial. Thus, $N\le {\sigma }^{-1}(N)$. This forces that $\sigma (N)\le N$, as desired.