Auto-invariance 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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Definition

Definition with symbols

A subgroup property $p$ is termed an auto-invariance property if there is a group-closed automorphism property $q$ such that for any subgroup $H \le G$, $H$ has property $p$ in $G$, iff any automorphism of $G$ satisfying property $p$, sends $H$ to within itself.

In terms of the function restriction formalism

A subgroup property $p$ is termed an auto-invariance property if there is a group-closed automorphism property $q$, such that we can write the following function restriction expression for $p$:

$p = q \to$ Function

In other words, a subgroup has subgroup property $p$ if every automorphism with property $q$, for the whole group, restricts to a function from the subgroup to itself.

This is equivalent to the following function restriction expressions:

$p = q \to$ Automorphism

and:

$p = q \to$ Endomorphism

In other words, the restriction is automatically guaranteed to be an automorphism of the subgroup.

Equivalence of definitions

The equivalence of definitions follows from the elementary observation: restriction of automorphism to subgroup invariant under it and its inverse is automorphism.

Examples

Normal subgroups

Further information: normal subgroup

The property of normality is an auto-invariance property, where the group-closed automorphism property in question is the property of being an inner automorphism.

Characteristic subgroups

Further information: characteristic subgroup

The property of being characteristic is an auto-invariance property, where the group-closed automorphism property in question is the property of being any automorphism.

Relation with other metaproperties

Weaker metaproperties

• Endo-invariance property
• Invariance property
• Strongly join-closed subgroup property
• Strongly intersection-closed subgroup property
• Automorphism-based relation-implication-expressible subgroup property
• Normalizer-closed subgroup property
• Centralizer-closed subgroup property
• Commutator-closed subgroup property