Self-centralizing and minimal normal implies 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., self-centralizing minimal normal subgroup) must also satisfy the second subgroup property (i.e., strictly characteristic subgroup)
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Statement

Verbal statement

Any 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.

Definitions used

Minimal normal subgroup

Further information: 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

Further information: self-centralizing subgroup

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

Strictly characteristic subgroup

Further information: 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. Self-centralizing and minimal normal implies monolith: In other words, a self-centralizing minimal normal subgroup is contained in every nontrivial normal subgroup.
2. 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.