Self-centralizing and minimal normal implies strictly characteristic

Verbal statement
Any fact about::minimal normal subgroup that is also self-centralizing (i.e., contains its centralizer in the whole group) is strictly characteristic: it is invariant under any surjective endomorphism of the group.

Minimal normal subgroup
A minimal normal subgroup of a group is a nontrivial normal subgroup that does not properly contain any other nontrivial normal subgroup.

Self-centralizing subgroup
A subgroup of a group is termed self-centralizing if it contains its centralizer in the whole group.

Strictly characteristic subgroup
A subgroup $$H$$ of a group $$G$$ is termed strictly characteristic if for every surjective endomorphism $$\sigma$$ of $$G$$, $$\sigma(H) \le H$$.

Related facts

 * Self-centralizing and minimal normal implies monolith
 * Self-centralizing and minimal normal implies characteristic

Facts used

 * 1) uses::Self-centralizing and minimal normal implies monolith: In other words, a self-centralizing minimal normal subgroup is contained in every nontrivial normal subgroup.
 * 2) uses::Monolith is strictly characteristic: A subgroup contained in every nontrivial normal subgroup is strictly characteristic.

Proof
Given: A group $$G$$, a minimal normal subgroup $$N$$ such that $$C_G(N) \le N$$. A surjective endomorphism $$\sigma$$ of $$G$$.

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

Proof: Consider the subgroup $$\sigma^{-1}(N)$$. This is normal by fact (2), so 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.