Left-inner implies intermediate subgroup condition
This article gives the statement and possibly, proof, of an implication relation between two subgroup metaproperties. That is, it states that every subgroup satisfying the first subgroup metaproperty (i.e., Left-inner subgroup property (?)) must also satisfy the second subgroup metaproperty (i.e., Intermediate subgroup condition (?))
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Left-inner subgroup property
Further information: left-inner subgroup property
A subgroup property is termed left-inner if there exists a property of functions from a group to itself such that can be written using the function restriction expression:
In other words, a subgroup of a group satisfies property in if and only if every inner automorphism of restricts to a function from to itself that satisfies .
Intermediate subgroup condition
Further information: intermediate subgroup condition
A subgroup property is said to satisfy the intermediate subgroup condition if, for any groups such that satisfies in , also satisfies in .
- Inner is extensibility-stable: An inner automorphism of a subgroup can be extended to an inner automorphism of the whole group.
- Left-extensibility-stable implies intermediate subgroup condition
The proof follows by combining facts (1) and (2).