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

Converse

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.