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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Minimal normal subgroupPLACEHOLDER FOR INFORMATION TO BE FILLED IN: [SHOW MORE]
Characteristically simple groupPLACEHOLDER FOR INFORMATION TO BE FILLED IN: [SHOW MORE]
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
- 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).
- Characteristic of normal implies normal: Every characteristic subgroup of a normal subgroup is normal in the whole group.
Given: is a minimal normal subgroup of a group .
To prove: is characteristically simple.
Proof: Suppose is a characteristic subgroup of . Then, by fact (1), is normal in . But since was minimal normal, either or is trivial. Thus, every characteristic subgroup of is either the whole of or is trivial.