Minimal normal implies characteristically simple

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

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.

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.