Potentially characteristic implies normal
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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 must also satisfy the second subgroup property
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Potentially characteristic subgroup
Further information: Potentially characteristic subgroup
A subgroup of a group is termed potentially characteristic if there exists a group containing , such that is a characteristic subgroup inside .
We use two facts in the proof:
- Every characteristic subgroup is normal
- Normality satisfies intermediate subgroup condition: If a subgroup is normal in the whole group, then it is normal in every intermediate subgroup.
Given: A group , and a potentially characteristic subgroup of
To prove: is a normal subgroup of
Proof: By the definition of potentially characteristic, there exists a group containing such that is characteristic inside .
Since every characteristic subgroup is normal, is a normal subgroup of .
Since , and normality satisfies intermediate subgroup condition, is also normal in . This completes the proof.