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

Related facts

• Monolith is fully characteristic in finite groups

Facts used

• 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 $N\leq M$ for any nontrivial normal subgroup $M$. A surjective endomorphism $\sigma$ of $G$.

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

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