Monolith is characteristic

Verbal statement
If a group has a monolith (a fact about::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).

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.