Minimal normal implies characteristically simple

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This fact is an application of the following pivotal fact/result/idea: characteristic of normal implies normal
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Statement

Verbal statement

Any group that occurs as a minimal normal subgroup of some group is characteristically simple.

Definitions

Minimal normal subgroup

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Characteristically simple group

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Related facts

Similar facts for specific kinds of groups

• Minimal normal implies additive group of a field in solvable
• Minimal normal implies elementary Abelian in finite solvable
• Minimal normal implies contained in Omega-1 of center for nilpotent p-group
• Minimal normal implies central in nilpotent

Converse

• Characteristically simple equals left-realization of minimal normal: Every characteristically simple group can be embedded as a minimal normal subgroup inside some group (in fact, inside its holomorph).

Facts used

1. Characteristic of normal implies normal: Every characteristic subgroup of a normal subgroup is normal in the whole group.

Proof

Hands-on proof

Given: $H$ is a minimal normal subgroup of a group $G$.

To prove: $H$ is characteristically simple.

Proof: Suppose $K$ is a characteristic subgroup of $H$. Then, by fact (1), $K$ is normal in $G$. But since $H$ was minimal normal, either $K=H$ or $K$ is trivial. Thus, every characteristic subgroup of $H$ is either the whole of $H$ or is trivial.