Left-inner implies intermediate subgroup condition

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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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Any left-inner subgroup property satisfies the intermediate subgroup condition.

Definitions used

Left-inner subgroup property

Further information: left-inner subgroup property

A subgroup property p is termed left-inner if there exists a property \alpha of functions from a group to itself such that p can be written using the function restriction expression:

inner automorphism \to \alpha

In other words, a subgroup H of a group G satisfies property p in G if and only if every inner automorphism of G restricts to a function from H to itself that satisfies \alpha.

Intermediate subgroup condition

Further information: intermediate subgroup condition

A subgroup property p is said to satisfy the intermediate subgroup condition if, for any groups H \le K \le G such that H satisfies p in G, H also satisfies p in K.

Facts used

  1. Inner is extensibility-stable: An inner automorphism of a subgroup can be extended to an inner automorphism of the whole group.
  2. Left-extensibility-stable implies intermediate subgroup condition


The proof follows by combining facts (1) and (2).