Left-inner subgroup property

Symbol-free definition
A subgroup property is said to be left-inner if, in the function restriction formalism, it has a restriction formal expression with the left side being inner automorphisms.

Definition with symbols
A subgroup property $$p$$ is said to be left-inner if, in the function restriction formalism, there exists a restriction formal expression for $$p$$ of the form:

Inner automorphism $$ \to b$$

where $$b$$ is any function property.

In other words, a subgroup $$H$$ satisfies property $$p$$ in $$G$$ if and only if every inner automorphism on $$G$$ restricts to a function on the subgroup satisfying property $$b$$.

In terms of the left expressibility operator
The subgroup metaproperty of being left-inner is obtained by applying the left expressibility operator to the function metaproperty of being exactly equal to the inner automorphism

Stronger metaproperties

 * Weaker than::Left-inner right-monoidal subgroup property

Weaker metaproperties

 * stronger than::Left-extensibility-stable subgroup property
 * stronger than::intermediate subgroup condition