Left-inner right-monoidal subgroup property

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This article defines a subgroup metaproperty: a property that can be evaluated to true/false for any subgroup property
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Statement

A subgroup property $p$ is termed a left-inner right-monoidal subgroup property if it can be expressed using a function restriction expression of the form:

Inner automorphism $\to b$

where $b$ is a monoidal function property -- in other words, for any group, the collection of functions from the group to itself satisfying property $b$ forms a monoid under composition.

Relation with other metaproperties

Weaker metaproperties

• Upper join-closed subgroup property: For full proof, refer: Left-inner right-monoidal implies upper join-closed
• Left-inner subgroup property
• Left-extensibility-stable subgroup property
• Intermediate subgroup condition