# Left-inner implies intermediate subgroup condition

From Groupprops

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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## Contents

## Statement

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 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 .

## Facts used

- 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

## Proof

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