Monolith is strictly 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 strictly characteristic subgroup (it is invariant under any surjective endomorphism of the group).

Related facts

 * Monolith is fully characteristic in finite groups

Facts used

 * uses::Normality satisfies inverse image condition: The inverse image of a normal subgroup under any homomorphism is normal.

Proof
Given: A group $$G$$, a minimal normal subgroup $$N$$ such that $$N \le M$$ for any nontrivial normal subgroup $$M$$. A surjective endomorphism $$\sigma$$ 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.