# Left-inner subgroup property

From Groupprops

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

View a complete list of subgroup metaproperties

View subgroup properties satisfying this metaproperty| View subgroup properties dissatisfying this metapropertyVIEW RELATED: subgroup metaproperty satisfactions| subgroup metaproperty dissatisfactions

## Definition

### 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 is said to be left-inner if, in the function restriction formalism, there exists a restriction formal expression for of the form:

where is any function property.

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

### 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